The Analytical Theory of Heat Jean Baptiste Joseph Fourier

Prévia do material em texto

; -. 
THE 
THEORY OF HEAT 
BY 
-J,~I] ]~I'r.d. JOSEPH FOURIER. 
TRANSLATED, WITH NOTES, 
BY 
ALEXANDER FREEMAN, M.A., 
FJIlLLOW O. IT JOHN'S COLLEOB, CAJilBBIDOB. 
BDITBD FOB THB SYNDICS OF THB UNIVBRSITY PBB88. 
Chmhdbgt: 
AT THE UNIVERSITY PRESS. 
LONDON: OAHBRIDGE WAREHOUSE, 17, PATERNOSTEB BOW. 
CAMBRIDGE: DEIGHTON. BELL, AND CO. 
LEIPZIG: F. A. ~ROCKBAUS. 
1878 
[AU Rightl re"rve.d.] ., 
Digitized by Coogle 
PRINTED BY C. J. CLAY. • ••• , 
AT TO VNIVBJI8ITY "'88. 
Digitized by Coogle 
) 
PREFACE. 
-
..... 
\;; 
" IN preparing this version m English of Fourier's 
.j 
... 
celebrated treatise on Heat, the tmnslator has followed 
faithfully the French original. He has, however, ap-
pended brief foot-notes, in which will be found references 
to other writings of Fourier and modern authors on 
the subject: these are distinguished by the initials A. F. 
The notes marked R. L. E. are taken from pencil me-
momnda on the margin of a copy of the work that 
formerly belonged to the late Robert Leslie Ellis, 
Fellow of Trinity College, and is now in the pOBBession 
of St John's College. It was the translator's hope to 
have been able to prefix to this treatise a Memoir 
of Fourier's life with BOme account of his writings; 
unforeseen circumstances have however prevented its 
completion in time to appeal" with the present work. 
r 
Digitized by Coogle 
• 
TABL~ 
CONTENTS OF THE WORKl. 
IIBBLIJ01IA&Y DIBCOtnISE • 
CHAPTER I. 
Introduction. 
SECTION I. 
SUDan o~ TIm OBnIcr o~ TBlI Woo • 
.lBT. 
PAOli 
1 
1. ObJ~ of the theoretical researches • 14 
2-10. Different uamplea, ring, cube, sphere, indnite prism; the variable 
temperature at any point whatever is a function of the coordinates 
and of the time. The quantity of heat, which during mlit of time 
crosses a given II1Jrlaoe in the interior of the solid, is also a function 
of the time elapsed. and of quantities which determine the form and 
position of the surface. The obj~ of the theory is to discover these 
hDctiODB 15 
11. The three lJISCiiic elements which m11lt be observed, are the capacity, the 
conducibility proper or pmrteabilitll. and the extemal conducibility or 
perutTabiUty. The codicients which express them may be regarded at 
fI1at .. CODBtaDt numbers, independent of the temperatures • 19 
11. Firat statement of the problem of the terrestrial temperatures 20 
13-15. Conditions neoeBSary to applications of the theory. Object of the 
eXpenments 21 
16-21. The rays of heat which escape from the same point of a 'surface 
have not the same intensity. The intensity of each ray is proportional 
1 Each parasr&ph of the Table indicates the matter veated of in the articles 
iDdicated at the left of that paragraph. The first of these articles begins at 
the psse marked on the right. 
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vi TABLE OF CONTENTS. 
ABT. '.LGB 
to the cosine of the ·angle which its direction makes with the normal to 
the 1I1Irfaoe. Divers remarks, and considerationa on the objeci and extent 
of thermological problema, and on the relations of general analysis with 
the study of nature • 22 
SECTION n. 
G& •• B.LL NOTloKa &aD PBBLllIllIABl' DUI1IITIOKa. 
22-24. Permanent temperature, thermometer. The temperature denoted 
by 0 Is that of melting ioe. The temperature of water boiling in a 
given veasel under a given preaB1ll'8 is denoted by 1 • !l6 
25. The unit which serves to measure quantities of heat, ia the heat 
required to liquify a certain mass of ioe . 27 
26. Specific capacity for heat • ib. 
117-29. Temperatures meaaured by increments of volume or by the addi· 
tional quantities of heat. Those cases only are here considered, in which 
the increments of volume are proportional to the increments of the 
quantity of heat. This condition does not in general emt in liquids ; 
it is sensibly true for solid bodies whose temperatures di1fer very much 
from those which cause the change of atate 28 
SO. Notion of external conducibility ib. 
81. We may at first regard the quantity of heat lost as proportional to the 
temperature. This proposition is not BCnsibly true except for oerlain 
limits of temperature 29 
82-85. The heat lost into the medium consists of several parts. The effeci 
is compound and variable. Luminous heat ib. 
86. Measure of the external conducibility 81 
87. Notion of the conducibility proper. This property alao may be observed 
in liquids. ib. 
SS, 89. Equilibrium of temperatures. The etJect ia independent of contact. 82 
4049. Firat notiona of radiant heat, and of the equilibrium which Is 
established in spaces void of air; of the cause of the reflection of rays 
of heat, or of their retention in bodies; of the mods of communication 
between the internal molecules; of the law which regulate. the inten· 
sity of the rays emiUed. The law is not disturbed by the reflection of 
heat. ib. 
50, 51. Firat notion of the effects of reflected heat 87 
52-56. Remarks on the statical or dynamical properties of heal It i. the 
principle of elasticity. The elastic forea of aeriform fluids exactiy indio 
C4tes their temperatures 89 
SECTION m. 
PBIlIOIPLB OJ' TIIB COJlJroKIC.LTlOH OJ' B&.a.T. 
57-59. When two molecules of the same solid are enremeJ,y near and at 
unequal temperatures, the most heated molecule communicates to that 
which is less heated a quantity of heat exaotly expressed by the product 
of the duraUon of the instant, of the extremely small dillerenoe of the 
temperatures, and of • certain funotion of the distance of the molecules. n 
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TABLE OF CONTENTS. vii 
All1'. NoOB 
60. When a heated body is plaoed in an aeriform medium at a lower tem· 
pera\ure, it loaea at each instant a quantity of heat which may be 
regarded in the first researches as proportional to the exce88 of the 
temperature of the surface over the temperature of the medium 48 
61-M. The propositions enunciaWi in the two preceding articles are founded 
on divers observations. The primary object of the theory is to disoover 
all the euct consequences of these propositions. We can then measure 
the variations of the aoefficieniB, by comparing the results of calculation 
with very exact ezperiments ill. 
SECTION IV. 
01' TBB UNll'OBM AND LlNBAB HOVllnNt' 01' HBA.1'. 
65. The permanent temperatures of an infinite solid included between two 
parallel planes maintained at fixed temperatures, are expressed by the 
equation <" - 0) e = (b - a) z; a and II are the temperatures of the two 
eztreme planes, e their distance, and II the temperature of the section, 
whose distance from the lower plarle is z • 45 
66, 67. Notion and measure of the fiow of heat 48 
68, 69. Heasure of the oonducibility proper . 61 
70. Remarks on the case in which the direct action of the heat extends to 
a aemdble distance • 68 
71. Sate of the same solid when the upper plane is exposed to the air ib. 
72. General conditions of the linear movement of heat 66 
SECTION V. 
LAW 01' THE PBBJUNBNr TIIIJlI'BBA1'UBBS iii! A PBISK 01' SIL\LL TBIODBSS. 
73-80. Equation of the linear movement of heat in the priam. Different 
consequences of this equation . 66 
SECTION VI. 
THE HBArING 01' CLOSED SUOKS. 
81-&. The final state of the solid boundary which encloses the space 
heated by a surface lI, maintained at the temperature II, is expressed by 
the foOowing equation: 
p 
m-n=(II-n) I+P' 
The value of P is ~ (~ + ~ +h-) , m is the temperature of the internal 
air, "the teDlperaiure of the external air, g, 11, H measure respectively 
the penetrability of the heated surface 11', that of the inner surface of the 
boundarJ " and u.t of the external surface,; e is the thickness of the 
boundary, and Kits conducibility proper • 62 
86, 86. Remarkable consequences of the precedingequation . 65 
87-91. Heasure of the quantity of heat requisite to retain at a constant 
temperature a body whose surface is protected from the external air by 
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viii TABLE OF CONT~ 
ABr. PAGB 
severa1l111C08118ive envelopes. Bemarkable eft_s of Ute aeparaUon of the 
81Ufaoes. neae resultll applioable &0 IIUU11 cWr_i pmblema 67 
SECTION VII. 
01' TD UlUI'OB. Ho'9BJBlll'r 01' Blur IX TJIlUIII DUlBX8IONS. 
92, 98. The permanent temperatures of a BOUd enclosed between m reo-
iaDgular planes are expressed by Ule equation 
tI=.A +az+bf/+C.l. 
1:, " • are Ute coordinates of any point, whose temperature is tI; .A, II, 
b, c are constant numbers. II Ule enreme pJanes are maintained by any 
causes at bed temperatures which satisfy the precediDg equation, the 
flDal syRem of aU the intemal temperatures will be expressed by the 
same equaUon • 78 
94, 95. JIeuure of Ute 1l0w of heat in this prism 75 
SEOTION VIn. 
Jb.uIUD 01' 'l'HB HOD .. n 01' Blur Ar A GIVIIK POIn 01' A GIVIIK SoLID. 
96-99. The variable system of temperatures of a solid is supposed &0 be 
expressed by the equation tI=F (II:, " I, t), where" denotes the variable 
temperature which would be observed after the time t had elapsed, at the 
point whose coordinates are 11:, 11,.. Formation of the analytical expres· 
Bion of the llow of heat in a given direotioD within the BOUd • 78 
100. Application of the preoediDg theorem &0 the aaae in which the fuDotion 
F is ,-.. COU: COl , co.. • 82 
CHAPTER II. 
Equation of the M 0tJement of Heat. 
SECTION I. 
EQUATION 01' TD VABmD HOU .. Nr 01' lbAT IN A BING. 
101-105. The variable movement of heat in a ring il expressed by the 
equation 
elf) K elt" III 
Iii = CD tk' - ODS f). 
The arc II: measures the distance of a section from the origin 0; f) is 
the temperature which that seotion acquires after the lapse of the time t; 
X, 0, D, " are the speci1lc coe1Iicienta; 8 is the area of the seotioD, by 
the revolution of which the riDg is genereted; Z is the perimeter of 
the eeotion 85 
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TABLE OF CONTENTS. ix, 
All'I'. PAOli 
106-110. The temperatures aipoints situated at equal distances are 
represented by the terms of a recurring series. Observation of the 
temperatures "l, "I' "a of three consecutive points gives the measure 
. A "+". " 8(108w)1 of the ratio g: we have ~ =q, ""-qw+I=O, and It =, Aloge • 
The distanoe between two oonsecutive points is~, and log w is the decimal 
logarithm of one of the two values of " 86 
SEOTION II. 
EOVATlOX 01' TD V ABIBD MOVJIJIBXT 01' IlBAT I. A SoLID BpD ••• 
111-118. ill denoting the radius of any shell, the movement of heat in the 
sphere is expreased by the equation 
~ = ~ (dip + ! "W) 
dI CD tk' ztk ." ." 
1U-l17. Conditions relative to the state of the 8Urlace and to the initial 
90 
atate of the solid 99 
SEOTION m. 
EOVATlOB 01' TlIB VAlullD'MoYBlIIBNT ol"HBAT IX :... BoLm OYLIBDBB. 
118-120. The temperatures of the solid are determined by three equations; 
the fim relates to the internal temperatures, the second expresses thO' 
eontinllOUB ltate of the surfaee, the third expl8l8es the initial atate of 
the solid • 95 
SEOTION IV. 
EOVATJOJlB 01' TlIB V ABIBD MovzJIZKT 01' HBAT Ill' A SoLID hJD 
01' IxnxITII LBXGTB. 
191-128. The system of fixed temperatures satisfies the equation 
"Iv II'" d'" 
.,.+ dy'+ .. =0; 
"is the temperature at a point whose coordinates are:li, y,' 97 
12., 125. Equation relative to the state of the surface and to that of the 
first section 99 
SEOTION V. 
EQVATlOBS OP TlIB V ABJKD MOVBJIBNT 01' HEAT IN A SoLID OVB •• 
126-181. The system of variable temperatures is determined by three 
equations; one expresses the internal sate, the seClond relates to the 
state of·the surfaee, and the third expresses the initial state • 101 
~a b 
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TABLE OF CONTENTS. 
SECTION VI. 
OF SoLIDS. 
ART. PAGB 
182-189. Elementary proof 'of properties of the uniform movement of heat 
in a solid enolosed betw~ six orthogonal planes, the CODBtant tem· 
peratures being expreBBed by the linear equation, 
,,=A -GIII-br-cz. 
The temperatures cannot change, since each point of the solid receives 
as much heat as it gives off. The quantity of heat which during the 
unit of time oroaaea a plane at right angles to the axis of I is the BaJDe, 
through whatever point of that aDs the plane paBll8S. The value of this 
common flow is that which would niBt, if the ooe1Jioients II and 6 
were nul • 104 
1'0, In. Analytical expreBBion of the flow in the interior of any solid. The 
equation of the temperatures being ,,= J(z, 11, I, t) the function - Kw ~ 
expresaea the qUantity of heat which during the instant dt croBaea an 
infinitely small area w perpendioular to the axis of I, at the point whose 
coordinates are :r, Vi I, and whose temperature is " after the time , 
baa e1epaed 109 
142-145. It is easy to derive from the foregoing theorem the general 
equation of the movement of heat, namely 
d" K (d'" d'" d"') iii = OJ) rl:c' + fill' + th' ... (A). • 112 
SECTION VII. 
GBNBRAL EQUATION RBL.\TIVII ro TJIII S11BI'A.CB. 
146-154. It is proved that the variable temperatures at points on the 
surface of a body, which is oooling in air, satisfy the equation 
d" d" dtI h 
"'di +"ilN +P a. + gtHl=o; md<i:+tad,+pda=O, 
being the differential equation of the surface which bounds the solid, 
and q being equal to (mt+Rt+p')~. To diBoover this equation we 
CODBider a molecule of the envelop which bounds the solid, and we express 
the fact that the temperature of this element does not ohange by a finite 
magnitude during an infinitely small iDBtant. This condition holds and 
continues to exist after that the regular action of the medium baa been 
exerted during a very small instant. Any form may be given to the 
element of the envelop. The case in which the molecule is formed by 
reotangula'r aeotiona presents remarkable properties. In the moat simple 
case, which is that in which the base is parallel to the tangent plane, 
the truth of ~e equation is evident • • 115 
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TABLE OF CONTENTS. Xl 
SECTION vnI. 
APPLICATION 01' TIIB GB1I&BAL EQUATIONS. 
ABT. '.lOB 
156, 156. In applying the general equation (A) to the case of the eylinder 
and of the iphere, we find "the same equations as those of Section DI. 
and of Section II. of thie chap_ 123 
SECTION IX. 
GB1IBBAL B:a1UBU. 
157-162. "Func1amental considerations on the DAture of the quantities 
Z, t, II, K. h, 0, D, whioh enter into all the analytical expressions of the 
Theory of Heat. Each of these quantities has an exponent of dimension 
which relates to the length, or to the duration, or to the temperature. 
These exponents are found by makin8 the units of measure vary. 126 
CHAPTER III. 
Propagation of Heat in an infinite rectangular IOlid. 
SECTION I. 
163-166. The constant temperaturel of a reota.ngu1a.r plate included be-
tween two parallel infinite sides. maintained at the temperature 0, aro 
• dw" dw" 
expressed by the equation dzI + dg" =0 • lSI 
167-170. If we considor tho state of the plate at a very great distance from 
the transverse edge, the ratio of the temperatures of two points whose 
ooordinakB are 1I:t, 71 and ~,,I changes according II the value of 1/ 
inor8B888; Zt and ; preserving their respeotive values. The ratio has 
a limit to which it approaches more and more, and when 11 is infinite, 
it is expressed by the product of a funotion of z and of a function of ,. 
This remark II1IfJloea to disclose the general form of ". namely, 
",.e -IJI-U.. (2i 1) II=~'-l ",e • 008 - .y. 
It is easy to ascertain how the movement of heat in the plate is 
effected, 134 
b2 
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ABT. 
TABLE OF CONTENTS. 
SECTIONn. 
PmaT EUJlPL. OJ' orn Villi: OJ' TmOONOJD:'rJIlO S.BDl:a nc 'rD 
TnoBY OJ' luu. 
171-178. Investigation of the ooe!icients in the equation 
l=a co8Z+i008 k+eooa 6e+clooa 7z+eto. 
From whioh we conclude 
40=_1_ ~ (-I)ttl 
... 1(-1.. ' 
.40B 
or 
.. 1 1 1 
,=00"-jOO8&1+600a&l-70087z+.. • • 187 
SEOTION m. 
n.Jl4JlK8 ON '1'BBaB BDDlB. 
179-181. To find the value of the series which forma \he seooud member, 
the number til. of terms is supposed to be llinited, and the series becomes 
a function of II: and fII. This function is developed according to powers of 
the reciproaa1 of fII, and fII is made infinite • 1"6 
182-184. The same proceBB is applied to several other series • 14.7 
185-188. In the- preceding development, which gives the value of the 
function of II: and tII., we determine rigorously the limits within which the 
sum of all the terms is included, starting from a given term • 
189. Very simple proceBB for forming \he series 
160 
.. i ... (-1)' 
i=- ~c- Ii -1 cos (2i -1) z. • • 168 
BEOTION IV. 
190, 191. Analytioal npreaaion 01 the movement of heat in • reetangular 
slab; it is decomposed into simple movements 154 
192-196. Measure of the quantity of heat which croBBe8 an edge or Bide 
parallel or perpendioular to the base. This expression of the flow II1lflicea 
to verify the solution • 166 
196-199. Oonsequences of this solution. The rectangular slab must be 
considered as forming part of an inflnite plane; \he solution expresses 
the permanent temperatures at all points of this plane • • 169 
200-204. It is proved that the problem proposed admits of no other 801u. 
tion dUfertln\ froll) tha~ which we have jUAt IItated • • 161 
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TABLE OF CONTENTS. 
SECTION V. 
FIlIl'nI EUUS810ll OP '1'IIlI B&St7IIr OP '1'IIlI SOL'O'rIOll. 
Aar. p~o. 
200, 206. The temperature at a point of the rectangalar slab whose co· 
ordiDahla are z and 110 is expreued thus 
• • 
SECTION VI. 
• 166 
DlmlLOPIllllft OP All AlmrrBABY FnO'l'lOll III TBlOOllOJIBTBIO BBBlBI. 
SO'l-214. The development obtained by determining the values of the un-
known coe1JioielltB iD the following equations idDite ill number ~ 
.A =G+2b+Sc+U+&o., 
B=G+2I b+8'e+4Itd+&O., 
O=G+II'6 + 81c + 4'4 + &0., 
D=G+lI7b+Blc+4f 4+&O., 
&0., &0. 
To 801ve theae equations, we first suppose the number of equations to be 
... and thai the number of unknowns G, b, c, cl, &0. is m only, omitting 
all the subsequent terms. The unknowns are determined for a oertain 
value of the number ... and ,he limits to which the values of the coeftl· 
cients continually approach are sought; these limits are the quantities 
which it is required to determine. Expression of the values of G, b, C, eI, 
&0. when m is infinite • 168 
215,216. The funotion Ijl(z) developed u.der the form 
G sinz+b sin 2a:+c sin Bz+d sin 4:1:+&0., 
whioh is tlrst supposed to contain only odd powers of z • • 179 
217, 218. Ditlerent expression of the same development. Application to the 
function e" - c-or • 181 
219-2111. Any function whatever Ijl (z) may be developed under the form 
4J sin z + lis sin.2z+ IIa sin 8z + ... + lis sin iz + &0. 
The value of the genaraJ ooem.oient lie is ! r _Ijl (z) sin i4D. Whence we 
"J. 
derive the very simple theorem 
jl/l(z) ... sinll r. daljl(a.}sina.+sin2a: f.. daljl(a.} sin ta. + sin Bz ;:. daljl(a.} sinBa.+&o., 
whence ~ I/I(z}=Z sin ia: r daljl(a.) sin'" • 184 
• ...1 J o · 
m, 2IIS. Application of the theorem: from it is derived the remarkable 
lI8rles, 
• 188 
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xiv TABLE OF CONTENTS. 
AB'1'. I'AO. 
224, 296. 8e00nd theorem on the development of functiona in bigono. 
mebical series: 
Applications: from it we derive the remarkable aeries 
! ... siD':I:=!'- col!2z _ COBU _ COB 6~ -&0. 
4 I 1.B B.o 0.7 
226-230. The preoeding theorems are applicable to disoontinuous functions, 
and solve the problems which are based upon the analysis of Daniel 
Bernoulli in the problem of vibrating cords. The value of the Beries, 
•• 1'211'2 1. D_· ..... _ B ..... am ~ V8l'IDD II + 2 am veram II + ii lID .... v........ 11+ ..... , 
is i, if we atbibute to ~ a quantity (!l'e&ter than 0 and leaa than II; and 
the value of the aeriea is 0, if ~ is any quantity included between II and i'l'. 
Application to other remarkable examples; curved linea or s111'faces which 
190 
coincide in a pad of their course, and di1rer in all the other pads. • 193 
231-288. Any function whatever, F(~), may be developed in the form 
F( )-A + S "I cou+/Iw cos 2z+/Iw COB &1:+&0., ~ - 161 siD~+6. siD 2:1+6, Bin 8~+&o. 
Each of the coefiicients is a definite integral. We have in general 
f + .. 91rA = _ .. cbP(~), f +.. . ftc = tkP(~) cos .. 
-.. 
. 
and 
. + .. 
ft, = i .. dzP(z) siD "" 
We thuB form the general theorem, which is one of the chief elements of 
our analysis: 
. 2"'P(~)='Z"'+«O (COSiZ!+"IIaP(II) COB la+siD iz!+" daP(a) siD is), 
t--_ -.. -W' 
284. The values of F(~) which correspond to values of z included 
between -'I' and +'1' must be regarded as entirely arbitrary. We may 
199 
also choose any limits whatever for z • 204 
280. Divers remarks on the use of developments in bigonomebic aeries • 206 
SECTION VII. 
Al'PLlCA'l'IOK '1'0 'l'JIJI ACrtlAL PBOBLJIK. 
286, 287. Expression of the permanent temperature in the infinite ractangnlar 
Blab, the state of the VanBverse edge being represented by an arbitrary 
function . 209 
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CHAPTER'IV. 
Of th8 lineat· and flaM Movement of Heat in a ring. 
SECTION I. 
GUBBAr. SOLUTIO. OJ' TBB l'BOBLB •• 
AB"r. PAO. 
238-241. The variable mOTeDlent which we are considering is compoaec1 of 
simple movemen1L In each of these movements, 1he temperatures pre. 
serve their primiuve rauos, and decrease with the time, aa the ordinates " 
of a liDe whose equation is ,,=.11.,......,. Formation of the general ex. 
pression • 118 
242-244. Application to some remarkable examples. Different oonsequenoes 
of the solution • 118 
146, t46. The system of temperatures converges rapidly towards a regular 
and final state, expressed by the ftrst part of the integral. The sum of 
the temperatures of 1wo points diametrically opposed is then the same, 
whatever be the position of the diameter. It is equal to the mean tem. 
perature. In each simple movement, the circumference is divided by 
equidistant nodes. All these partial movements BUOOessively disappear, 
excep1 the fint; and in general the heat distnDuted throughout the solid 
asaumes a regular disposition, independent of the iniUal staw • III 
SECTION II. 
947-160. Of the commtmicauon of heat between two maBsel. Expression 
of the variable temperatuteB. Remark on the value of the ooeJ!ioient 
which measures the conduoibility • tt6 
151-165. Of the commtmication of hea1 between " separate maBBeB, ar. 
rauged in a straight line. Expression of the variable temperature of each 
mass j it is given as a function of the time elapsed, of the coefficient 
which measures the conduoibility, and of all the initial temperatures 
regarded lUI arbitrary • 128 
156, 257. Bemarkable consequences of this soluuOD • 236 
258. Application to the case in whioh the number of muses is infinite. • 137 
159-266. Of the commtmicauon of heat between R separate masses arranged 
circularly. Differential equations suitable to the problem j integration of 
these equations. The variable temperature of each of the masses is ex. 
pressed as a funotion of the coefficient which measures the conduoibility, 
of the time which has elapaed since the instaut when the oommunication 
bepn, and of all the initial temperatures, whioh are arbitrary; but in 
order to determine these funotions completely, it is necessary to ded 
the elimination of the ooefficients• 288 
267-271. Elimination of the coeffioients in the equationa which contain 
these unknown quantities and the given initial temperatures • • 247 
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xvi TABLE 01' CONTENTS. 
AlIT. PAO. 
272, 278. Formation of ille general aolution: aDalytioal expression of ille 
result 268 
274-276. AppUoation and coDBequenC8B of Urla aolution • 266 
277. 278. Examination of ille case in which ille number" is mpposed infinite. 
W$ obtain ille solution relative to a solid ring, set forill in Article 241, 
and ille illeorem of Article 284. We ilius aaoerlain ille origin of ilie 
analysis which we have employed to aolve ille equation relating to con· 
tinuous bodies • 269 
279. Analytical expression of ille two preceding results 2611 
1180-282. U is ~oved Ulat ille problem of the movement of heat in a ring 
admits no other solution. The integral of the equation :' = k: is 
evidently ille most general which can be formed 
CHAPTER V. 
Of tM Propagation of Heat in a solid sphere. 
SECTION L 
GB1'IBIUL BoL11TJ01'l. 
283-289. The ratio of the variable temperatures of two points in ille aolid 
is in ille first place considered to approach continually a definite limit. 
This remark leads to ille equation t1=,.{ sin ftZ e-ll.." which expreasetl 
:I: 
the simple movement of heat in ille sphere. The number " baa an 
in~nity. of values given by ille definite equation t":~x = I-AX. The 
radius of ille sphere is denoted by X, and ille radius of any conoentrio 
sphere, whose temperature is t1 after ille lapse of ille time '. by :1:; 1a 
and K are ille speoiJic coefticients; ,.{ is any constant. ConstruotioDB 
adapted to discloae ille nature of the definite equation. the limita and 
va!ues Ilf its ~ota. . . , 268 
290-2!12. Formation,of ill,e generalaolution; final state of ille aolid. • 274 
29S. Application to the oaae in which the sphere baa been heated by a ~ 
longed immersion 277 
SECTION IL 
DIn1Iiu1'I1' UKABU 01'1 TJlIS 8oL11TJ01'l. 
294-296. BemIts relative to spheres of BIDall radius, and to ille final tem· 
peratures of any sphere • • 279 
298-800. Variable tQm~ture of • thermometer plunged into a liquid 
whioh is cooling freely. Application of ille resuUa to ille comparison and 
111141 of tbermometera • 282 
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TABLE OF CONTENTS. xvii 
£ft. PAG. 
SOl. Espresaion of the mean kmperature of the sphere 81 a function of the 
time elapsed • 286 
~. Application to sph8l'8ll of 'tfJr1 great radius, and to thote in which 
ihe radius is very amaIl • 287 
SOli. :Remark on the nature ~f the de1lBi~ equaUon which giT88 all the valU88 
of". • . 289 
CHAPTER VI . 
. 01 tAe Movement of H~t ~n a 80lid cylinder. 
1106., 807. We remark in the !rat place that the ratio of the variable ~m. 
peratures of two pointe of the solid approaches oontinually a definite 
limU, and by this we aaoerlain the expreBBion of the aimple m~ement. 
The function of l1li which ~ one of the factors of this expression is given 
by a differential equation of the second order. A number 9 enters into 
this function, and must satisfy a detlnite equatioD. • • • • 291 
808, 809. An8Jysis of this equation. By means of the principal theorems of 
algebra, it is proved that all the roots of the equation are real • • 294 
810. The function" of the variable l1li is expressed by 
1 10" " =- if' GOB (zJi sin r); 
.. 0 
anil the definite equation is h+ ~ =0, giving to l1li its oomplete value X. 296 
811, 812. The development of the funotion fI (z) IMliDs represeatN by 
the value of the series 
is 
s' .. 
G+bz+Ci+ 1i 2.8 +&0., 
cP ,I' g'-
G+ji + 21.41+ 21.41.61+&0., 
.!. ~ dul,6(C ainu). 
.".Jo 
Remark on this UBI of definite integrals • 298 
818. Expression of the function u of the variable l1li as a oontinued fraetion. 800 
814. Formation of the general solution • 801 
815-818. Statement of the analysis which determines the values of tbe.oo-
elliciente • 303 
819. General solution 808 
820. COnaequen088 of the solution • 809 
Digitized by Coogle 
xviii TABLE OF CONTENTS. 
CHAPTER VII. 
. . . . . 
Pr(YiJO{JafMm, of Heat in a rectangular prism. 
ABT. .,. . P40. 
821-828. Expression of the Bimple movement determined by . the general 
p~per\ies 01 h8l\t, aJ\d by. the ~orm of tl\e eol,id. Jnto ~ oxprepion 
enters an arc • which satisfies a tranaoendental equation, all of whose 
roots are real • • 811 
824. All the unknown ooefticients are determined by de1lnite integrals • 818 
825. General solution of the problem • 814. 
826, 827. The problem proposed admits no other solution • • 815 
828, 829. Temperatures at points on the uis of the prism . 817 
880, Applioation to the case in which the thickness of the prism is very 
small • 818 
881. The solution shows how the uniform movement of heat is established 
in the interior of the solid 819 
882. Application. to prisms, the dimensions of whose bases are large. • 822 
CHAPTER VIII. 
Of the Movement of Heat in a solid cube. 
888, 8M. Expression of the Bimple movement. Into it enters an arc f 
whioh must satisfy a trigonometrio equation all of whose roote arc real • 828 
835, 336. Formation of the general solution • 824: 
887. The problem can admit no other solution 827 
88S. Consequence of the solution • ib. 
389. Expression of the mean temperature 828 
84:0. Comparison of the final movement of heat in the oube, with the 
movement which takes place in the sphere 829 
In. Applioatioa to the simple case OODeidered in An. 100 881 
CHAPTER IX. 
Of the Diffusion of Heat. 
SECTION I . 
. 01' TO I'.IID MoVBJIBu. 01' lIB4'l' IN All IxI'DIJ'l'II LINE. 
842--847. We consider the linear movement of heat in an infinite line, a 
part of which has been heated; the initial state is represented by 
11= F (:I:). The following theorem is proved: 
i' F(:I:) = i.e tbJ 008 q:l: t, da F{IJ) cos qIJ, 
Digitized by Coogle 
TABLE OF CONTENTS. xix 
AA'l'. PAOB 
The funetion P(z) aatia1lea the condition P(z)=P( -z). Espresaion of 
the variable temperatureB • 888 
848. Application to. the .oue • in which all the points of the pan heated 
have received the aame initial temparature. The iniegral 
rID dq Bin q COB P ill I .. , Jo q 
if we give to s a value included between 1 and - 1. 
The definite integral has a nul value, if • is no' included between 
land -1 • 
1U9. Application to the case in which the heating given results from the 
final Btate which the action of a BOurce of heat determineB 
SGO. Discontinuoua values of the function espre&Bed by the integral 
rID dq Jo l+ qI COB!lZ 
851-863. We CODBider the linear movement of heat in • line whose initial 
temperatureB are represented by fJ=f(z) at the distance .'to the right 
of the origin, and by fJ = - I (z) at the distance z to the left of the origin. 
EspreBBion of the variable temperature at any point. The solution 
derived from the analysis which expreBSeB the movement of heat in an 
888 
889 
infinite line ib. 
864. ExpreBBion of the variable tempera'urea when the initial state of the 
pari beated is expre&Bed by an entirely arbitrary function • 848 
855-358. The developmentB of functioDB in Bines or cosines of multiple area 
are transformed into definite integrals 846 
Si9. The following theorem is proved : 
.. /.ID lID '01(z) = dqBinqs ciAI(II) BinqeL 
.. .. .' 
The funetion I (z) aatiB1ieB the condition: 
f(-z)= -fez) • 348 
lJ6O.-.362. Use of the preoediDg results. Proof of the theorem expreaBed 
by the general equation: 
"(z)= J:ciA4><II) LID dqCOI(qZ-qll). 
ThiB equation ill evidently included in equation (II) stated in Art. 284. 
(See Art. 397) • ib. 
868. The foregoing solution Bbewl also the variable movement of beat in an 
infinite line, one point of which is Bubmitted to a conatant temperature • 852 
864. 'l'be lame probmm may also be solvedby meana of "aIlother form of the 
integral. Formation of this iniegral 854 
365, 866. Application of tbe solution to an infinite prism, whOle initial 
temperatures are nul. Bemarkable cODBequenceB • 856 
867-869. The same integral applieB to tbe problem of the diftuaion of beat. 
The solution whicb we derive from ii agrees with that which has been 
Btated in Articles 347,348 362 
Digitized by Coogle 
Xx TABLE OF CONTENTS. 
.lBT. n08 
870, 871. Remarks on di1Immt forms of the integral of the equation 
du diu 
dC=dzI • • S66 
SEOTION II. 
01' DII no 1Il0 .... JDlNT 01' HUT Dr .lK lNl'DIITB SoLm. 
872-876. The es:preaaion for the variable movement of heat in an infinite 
solid JDaIIB, aooording to three dimensions, is derived immediately from 
that of the linear m01'ement. The integral. of Che equation 
flll tPtI tPtI d'v 
CU = rkl + dyl + dzs 
solves the proposed problem. It cannot ha1'e a more extended integral ; 
it is deri1'ed also from the particular value 
or from this; 
... 
.,-4i 
.... ,Jc' 
which both satisry the equation :: = :.. The ~en~ty of the in-
tagrala obtained is follDded upon the following proposition, which may be 
regardod as. self-evident. Two functions of the variables z", .. C are 
neoessarily identical, if they satisfy the di1Ierential equation 
dtI dill dill flItI 
dt=dz2+fly,+dzl' 
and if at the same time they have the _saDle value for a certain value 
~c • m 
877-882. The heat contained in a part of an infinite prism, all the other 
pointe of which have nul initial temperature, begins to be distributed 
throughout the whole mass j and after a certain interval of time, the 
state of any part of the solid depends not upon the distribution of the 
initial heat, but simply upon its quantity. The last result is not due 
to the increase of the distance included between any point of the mass 
and the part which has been heated; it is entirely due to the increase 
of the time elapsed. In all problems submitted to analysis, the expo-
nents are absolute numbers, and not quantities. We ought not to omit 
the parte of these eXl'onents which !U"8 ~coml'arab!y 8~aller than the 
others, but only those whose absolute values are extremely small • 876 
888-885. The SaDle remarks apply to the distribution of heat in an infinite 
solid. • • 882 
SECTION m 
Tn HIOHBST TBKPBB.lTtl'BBS Dr .llII INPDIITB SOLlD. 
886,8137. The heat contained in part of the prism distributes itself through-
out the whole mass. The temperature at a distaut point rises pro-
gressively, arrives at its sreateat value, and then decreases. The time 
Digitized by Coogle 
TABLE OF CONTENTS. xxi 
AJrr. PAO. 
after which this muimum oooura, is a funotion of the distance lit. 
Expression of this fllnotion for a prism. whose heated points have reo 
ceived the same iniUal temperature • S86 
888-S91. Solution of a problem analogous to the foregoing. Difrel'8Jlt 
results of the solution 887 
892-896. The movement of heat in an infinite eolid is considered; and 
the highest temperatures. at parts very distant from the pari originally 
heated, are determined • 899 
SECTION IV. 
eo.PABleoR 01' TID buGBlLS. 
896. Him integral (el) of the equation ~ = :; (IJ). This integral e:rpreaaea 
the movement of heat in a ring • 896 
897. Second integral (fl) of the same equation (IJ). It expressea the linear 
mOVlllllent of heat in an infinite solid • 898 
898. Two other forms (-)0) and (&) of the integral. which are d~ved, like the 
preceding form. from the integral (el) • lb. 
899, 400. Firat development of the value of 11 aooording to increasing powers 
of the time f. Second development aoeording to the powers of fl. The 
first must contain a single arbitrary funotion of f • • 899 
401. Notation appropriate to the representation of these developments. The 
analysis which is derived from it dispenses with effeoting the develop. 
ment in aeries • • 402 
402. .Application to the equation. : 
d'tI d'tI dItI dItI de" 
dt' = ck'+ d,,··;···(C), and tJtI + fh:'=O ...... (d) • • 404 
408. Application to the equations: 
dItI de" de" d'" 
iUi + d.i:' + 2 US.d" + dg4=O ...... (e). 
and dfl dItI de" dfItI d' = IJ dzI t b .. + e Ik' + &c ...... (f) • 406 
404. Use of the theorem B of Article 861, to form the integral of equation (I) 
of the preceding Article 407 
406. Use of the same theorem to form the integral of equation (tl) which 
belongs to elastio platel 409 
406. Second form of the same integral • 412 
40'7. Lemmas whioh serve to effect these transformations 418 
408. The theorem expressed by equation (E). Art. 861, applies to any number 
of variables 416 
409. Use of this proposition to form the integral of equation (e) of An. 402. 416 
410. Application of the same theorem to the equation 
d'tI d'tI d'tI 
d:I:' + d,t + d:" = 0 • • 418 
Digitized by Coogle 
xxii TABLE OF CONTENTS. 
AlIT. PAO. 
411. Integral of equation (e) of vibrating elastio aurfaoeI 419 
412. Seoond form of the integral • 421 
418. Use of the same theorem to obtain the integrals, by II11JDJD.ing the 
l8liae which repI9II8Ilt them. Applioation to the equation 
Ii" d't1 
dt = del' 
IJ;I.tegr~ un.,er fil1ite form containing two arbitrary funotions of,. • 422 
414. The expreaaioDB change form when we use other limits of the definite 
integrals • US 
416, 416. Conatruotion which aenae to prove the senera1 equation 
11+- 1+-I(a:)=k __ da/(Ct) __ "'OOB~-pCt) ...... (B) 
417. Any limits IJ and b may be taken for the integral with respect to II. 
These limits are those of the "aluee of z which correepond to esisting 
values of the function I(z). Every other value of tIC gives a nul result 
ib. 
for I (z) • 429 
418. The same remark appUae to the general equation 
1 ,-+- 1+- 2ill" I(z) =211'2:,___ __ dll/(lI) cosx (z-Ct), 
the aeoond member of.which repreaents a periodio funotion "2 
419. The chief character of the theorem expreBBed by equation (B) conaists 
in this, that the·sign lof the function is transferred to another unknown 
m. and.that the oIlief 1Viable z ia only under the symbol ooeiDe 483 
420. Use of these theorems in the analysis of imaginary quantities • 4SO 
421. :ipp~cation to the equation ::; + :: = 0 486 
422. General expression of the fiuxion of the order i, 
rJ.I·/(z) (iT • 
42S. Construction which senes to prove the general equation. Consequences 
.tiVV to ~e ~nt of equations of this kind, to the values of 1(<<) 
which correspond to the limits of lie, to the infinite values of 1(1Ie) • 
424-427. The method whioh Mnsists in determining by definite integrals 
the unknown coefficients of the development of a funotion of z under 
the form 
a~{I'J.~ + 1lt;(~1 +~W) +&0., 
is deriTed from the elements of algebraio analysis. Example relative to 
the distribution of heat in a soUd sphere. By examining from this 
point of view the process which Benes to determine the coefficients. we 
solve easily problems which may arise on the employment of all the terms 
of the second member, on the discontinuity of functions, on singular or 
infinite values •. The equations which are obtained by this method ex. 
press either the variable state, or the initial state of masses of infinite 
dimensions.' The! form of the integrals which belong to the theory of 
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"7 
TABLE OF CONTENTS. xxiii 
ART. ..&0. 
heat, represents at the same time the composition of simple movements, 
and that of an iDtinity of partial elfecta, due to the action of all points of 
the BOlid • 441 
~8. General remarks on the method which haa served to BOlve the analytical 
problema of the theory of heat • • 450 
~9. General remarks on the principles from which we have derived the dif· 
ferential equations of the movement of heat • 456 
480. Terminology relative to the general properties of heat • , 462 
431. Notationsproposed 463 
432, 438. General remarks on the nature of the eoefficients which enter into 
the clliIerential equations of the movement of heat , • 466 
EBBATA. 
Page 9, line lIS,/or m. read IV. 
Pages M, 65, for Ie read K. 
Page 189, line 2, The equation should be denoted (A). 
Page 206, last line but one, for #II read X. 
Page 298, line 18, for :: read ::. 
Page 299, line 16, for 01 read ,ft. 
.. .. lad line, read 
r. flu ~ (t siD u)= .. ~+tSl.;' + 5 SJ~" +&0. 
Page SOO, line 8, lor .A •• .A •• .A •• read .. .A •• .. .d., .. ..4 •• 
Page 40'7, line 12. for d1J read dp. 
Page 432, line 18, read (:1: - II). 
Digitized by Coogle 
Digitized by Coogle 
PRELIMINARY DISCOURSE. 
PnlllARY causes are unknown to US; but are subject to simple 
and constant laws, which may be discovered by observation, the 
study of them being the object of natural philosophy. 
Heat, like gravity, penetrates every substance of the universe, 
its rays occupy all parts of space. The object of our work is to 
set forth the mathematical laws which this element obeys. The 
theory of heat will hereafter form .one of the most important 
branches of general physics. 
1.'he knowledge of rational mechanics, which the most ancient 
nations had been able _to acquire, has not come down to us, and 
the history of this science, if we except the first theorems in 
harmony, is not traced up beyond the discoveries of Archimedes. 
This great geometer explained the mathematical principles of 
the equilibrium of solids and fluids. About eighteen centuries 
elapsed before Galileo, the originator of dynamical theories, dis-
covered the laws of motion of heavy bodies. Within this new 
science Newton comprised the whole system of the universe. The 
successors of these philosophers have extended these theories, and 
given them an admirable perfection: they have taught us that 
the most diverse phenomena are subject to a small number of 
fundamental laws which are reproduced in all the acts of nature. 
It is recognised that the same principles regulate all the move-
ments of the stars, their form, the inequalities of their courses, 
the equilibrium and the oscillations of the seas, the harmonic 
vjbrations of air and sonorous bodies, the transmission of light, 
capillary actions, the undulations of fluids, in fine the most com-
plex effects of all the natural forces, and thus has the thought 
~a 1 
Digitized by Coogle 
2 THEORY OF HEAT. 
of Newton been confirmed: quod. tam paw tam multa prtBBtet 
geometria glmiatur1• 
But whatever may be the range of mechanical theories. they 
do not apply to the effects of heat. These' make up a special 
order of phenomena, which cannot be explained by the principles 
'of motion and equilibrium. We have for a long time been in 
possession of ingenious instruments adapted to measure many 
of these effects; valuable observations have been collected; but 
in this manner partial results only have become known. and 
not the mathematical demonstration of the laws which include 
them all 
I have deduced these laws from prolonged study and at-
tentive comparison of the facts known up to this time: all these 
facts I have observed afresh in the course of several years with 
the most exact instruments that have hitherto been used. 
To found the theory, it was in the first place necessary to 
distinguish and define with precision the elementary properties 
which determine the action of heat. I then perceive,l that all the 
phenomena which depend on this action resolve themselves into 
a very small number of general and simple facts; whereby every 
physical problem of this kind is brought back to an investiga-
tion of mathematical analysis. From these general facts I have 
concluded that to determine numerically the most varied move-
ments of heat, it is sufficient to submit each substance to three 
fundamental observations. Different bodies in fact do not possess 
in the same degree ~e power to contain heat, to recsive or tra1l8mit 
it a.ct'OBB their IlUrfac6l. nor to conduct it through the interior of 
their masses. These are the three specific qualities which our 
theory clearly distinguishes and shews how to measure. 
It is easy to judge how much these researches concern the 
physical sciences and civil economy, and what may be their 
influence on the progress of the arts which require the employ-
ment and distribution of heat. They have also a necessary con-
nection with the system of the world. and their relations become 
known when we consider the grand phenomena which take place 
near the surface of the terrestrial globe. 
1 Philolophi<e fJ4turalil principia mathtmatica. .A uctoril prttJatio ad kctortffl. 
Ao gloriatur geometria quod tam paucia prinoipiil aliunde peUtia tam multa 
p1'llBtel [A. F.] 
Digitized by Coogle 
PRELIJUNARY DISCOURSE. 3 
In fact, the radiation of the sun in which this planet is 
incessantly plunged, penetrates the air, the earth, and the waters; 
its elements are divided, change in direction every way, and, 
penetrating the mass of the globe, would raise its mean tem-
perature more and more, if the heat. acquired were not exactly 
balanced by that which escapes in rays frOm all points of the 
surface and expands through the sky. 
Different climates, unequally exposed to the action of solar 
heat, have, after an immense time, acquired the temperatures 
proper to their situation. This effect is modified by several ac-
cessory causes, such as elevation, the form of the ground, the 
neighbourhood and extent of continents and seas, the state of the 
surface, the direction of the winds. 
The succession of day and night, the alternations of the 
seasons occasion in the solid earth periodic variations, which are 
repeated every day or every year: but these changes become 
less and less sensible as the point at which they are measnred 
recedes from the surface. No diurnal variation can be detected 
at the depth of about three metres [ten feet] j and the annual 
variations cease to be appreciable at a depth much less than 
sixty metres. The temperature at great depths is then sensibly 
fixed at a given place: but it is not the same at all points of the 
same meridian j in general it rises as the equator is approached. 
The heat which the sun bas communicated to the terrestrial 
globe, and which bas produced the diversity of climates, is now 
subject to a movement wbich bas become uniform. It advances 
within the interior of the mass which it penetrates throughout, 
and at the same time recedes from the plane of the equator, and 
proceeds to lose itself across the polar regions. 
In ~e higher regions of the atmosphere the air is very rare 
and transparent, and retains but a minute part of the heat of 
the solar rays: this is the cause of the excessive cold of elevated 
places.. The lower layers, denser and more heated by the land 
and water, expand and rise up: they are cooled by the very 
fact of expansion. The great movements of the air, such as 
the trade winds which blow between the tropics, are not de-
termined by the attractive forces of the moon and sun. The 
action of these celestial bodies produces scarcely perceptible 
oecillations in a fluid so rare and at 80 great a distance. It 
1-2 
Digitized by Coogle 
THEORY OF HEAT. 
is the changes of temperature which periodically displace every 
part of the atmosphere. 
The waters of the ocean are differently exposed at their 
surface to the rays of the sun, and the bottom of the basin 
which contains them is heated very unequally from the poles 
to the equator. These two causes, ever present, and combined 
with gravity and the centrifugal force, keep up vast movements 
in the intel'ior of the seas. They displace and mingle all the 
parts, and produce those general and regular currents. which 
navigatQrs have noticed. 
Radiant hentwhich escapes from the surface or all bodies, 
and travarsell elastic media, or spaces void of air, has special 
laws, and occurs with widely varied phenomena. The physical 
explanation of many of these facts is already known; the mathe-
matical theory which I have formed gives an exact measure of 
them. It consists, in a manner, in a new catoptrics which 
has its own theorems, and serves to determine by analysis all 
the effects of heat direct or reflected. 
The enumeration of the chief objects of the theory sufficiently 
shews the nature of the questions which I have proposed to 
myself. What are the elementary properties which it is requisite 
to observe in each substance, and what are the experiments 
most suitable to determine them exactly 1 If the distribution 
of heat in solid matter is regulated by constant laws, what is 
the mathematical expression of those laws, and by what analysis 
may we derive from this expression the complete solution of 
the principal problems 1 Why do terrestrial temperatures cease 
to be variable at a depth so small with respect to the radius 
of the earth? Every ineqUJility in the movement of this planet 
necessarily occasioning an oscillation of the solar heat beneath 
the surface, what relation is there between the duration of its 
period, and the depth at which the temperatures become con-
stant 1 
What time must have elapsed before the climates could acquire 
the different temperatures which they now maintain; and what 
are the different causes which can now vary their mean heat 1 
Wfty do not the annual changes alone in the distance of the 
sun from the earth, produce at the surface of the earth very 
considerable changes in the temperatures? 
Digitized by Coogle 
• 
PREJ.D[lNABY DISCOURSE. 
From what characteriRtic can we ascertain that the earth 
has not entirely lost its original heat; and what are the exact 
laws of the loss? 
If, as several observations indicate, this fundamental heat 
is not wholly dissipated, it must be immense at great depths, 
and nevertheless it has no sensible influence at tho present time 
on the mean temperature of the climates. The effects which 
are observed in them are due to the action of the solar rays. 
But independently of these two sourcE'S of hea.t, the one funda-
mental and primitive, proper to the terrestrial globe, the other due 
to the presence of the sun, is there not a. more universal cause, 
which determines the temperature of the hea'IJen8, in that part 
of space which the solar system now occupies! Since the ob-
served facts necessitate this cause, what are the consequences 
of an exact theory in this entirely new question; how shall we 
be able to determine that constant value of the temperature of 
, apace, and deduce from it the temperature which belongs to each 
planet? 
To these questions must be added others which depend on 
the properties of radiant heat. The physical cause of the re-
:8ection of cold, that is to say the reflection of a lesser degree 
of heat, is very distinctly known; but what is the mathematical 
expression of this effect? . 
On what general principles do the atmospheric temperatures 
depend, whether the thermometer which measures them receives 
the solar rays directly, on a surface metallic or unpolished, 
or whether this instrument remains exposed, during the night, 
under a sky free from clouds, to contact with the air, to radiation 
from terrestrial bodies, and to that from the most distant and 
coldest parts of the atmosphere 1 
The intensity of the rays which escape from a point on the 
surface of any heated body varying with their inclination ac-
cording to a law which experiments have indicated, is there not a 
necessary mathematical relation between this la.-w and the general 
fact of the equilibrium of heat; and what is the physical cause of 
this inequality in intensity? 
Lastly, when heat penetrates fluid masses, and determines in 
them internal movements by continual changes of the temperature 
"nd density of each molecule, can we -still express, by differential 
Digitized by Coogle 
6 THEORY OF HEAT. 
equations, the laws of such a compound effect; and what is the 
resulting change in the general equations of hydrodynamics , 
Such are the chief problems which I have solved, and which 
have never yet been submitted to calculation. If we consider 
further the manifold relations of this mathematical theory to 
civil uses and the technical arts, we shall recognize completely 
the extent of its applications. It is evident that it includes an 
entire series of distinct phenomena, and that the study of it 
cannot be omitted without losing a notable part of the science of 
nature. 
The principles of the theory are derived, as are those of 
. rational mechanics, from a very small number of primary facts, 
the causes of which are not considered by geometers, but which 
they admit as the results of common observations confirmed by all 
experiment. 
The differential equations of the propagation of heat express 
the most general conditions, and reduce the physical questions to 
problems of pure analysis, and this is the proper object of theory. 
They are not less rigorously established than the general equations 
of equilibrium and motion. In order to make this comparison 
more perceptible, we have always preferred demonstrations ana-
logous to those of the theorems which serve as the foundation 
of statics and dynamics. These equations still exist, but receive 
a different form, when they express the distribution of luminous 
heat in transparent bodies, or the movements which the changes 
of temperature and density occasion in the interior of fluids. 
The coefficients which they contain are subject to variations whose 
exact tileasllre is not yet known; but in all the natural problems 
which it most concerns us to consider, the limits of temperature 
differ so little that we may omit the variations of these co-
efficients. 
The equations of the movement of heat, like those which 
express the vibrations of sonorous bodies, or the ultimate oscilla-
tions of liquids, belong to one of the most recently discovered 
branches of analysis, which it is very important to perfect. After 
having established these differential equations their integrals must 
be obtained; this process consists in passing from a common 
expression to a particular solution subject to all the given con-
ditions. This difficult investigation requires a special analysis 
Digitized by Coogle 
PRELDlIJfABY DISCOURSE, 7 
founded on new theorems, whose object we could not in this 
place make known. The method which is derived from them 
]eaves nothing vague and indeterminate in the solutions, it leads 
them up to the final numerical applications; a necessary condition 
of every investigation, without which we should only arrive at 
useless transformations. 
The same theorems which have made known to us the 
equations of the moveinent of heat, apply directly to certain pro-
blems of general analysis and dynamics whose solution has for a 
long time been desired. 
Profound stud of nature is th 0 source of mathe-
matt discoveries. ot only has this study, in offering a de-
rermmate object to lUvestigation, the advantage of excluding 
vague questions and calculations without issue; it is besides a 
sure method of forming analysis itself, and of discovering the 
e]ements which it concerns us to know, and which natural science 
ought always to preserve: these are the fundamental elements 
which are reproduced in all natural effects. 
We see, for example, that the same expression whose abstract 
properties geometers had considered, and which in this respect 
belongs to general analysis, represents as well the motion of light 
in the atmosphere, as it determines the laws of diffusion of heat 
in solid matter, and enters into all the chiefproblems of the 
theory of probability. 
The analytical equations, unknown to the ancient geometers, 
which Descartes was the first to introduce into the study of curves 
and surfaces, are not restricted to the properties of figures, and to 
those properties which are the object of rational mechanics j they 
extend to all general phenomena. There cannot be a language 
more universal and more simple, more free from errors and from 
obscurities, that is to say more worthy to express the invariable 
relations of natural things. 
Considered from this point of view, mathematical analysis is as 
extensive as nature itself; it defines all perceptible relations, 
measures times, spaces, forces, temperatures j this difficult science 
is formed slowly, but it preserves every principle which it has once 
acquired j it grows and strengthens itself incessantly in the midst 
of the many variations and errors of the human mind. 
Its chief attribute is clearness j it has no marks to express con-
Digitized by Coogle 
8 THEORY OF HEAT. 
~ fused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them. If matter 
escapes us, as that of air and light, by its extreme tenuity, if 
bodies are placed far ·from us in the immensity of space, if man 
wishes to know the aspect of the heavens at successive epochs 
separated by a great number of centuries, if the actions of gravity 
and of heat are exerted in the interior of the earth at depths 
which will be always inaccessible, mathematical analysis can yet 
lay hold of the laws of these phenomena. It makes them present 
and measurable, and seems to be a faculty of the human mind 
destined to supplement the shortness of life and the imperfec-
tion of the senses; and what is still more remarkable, it follows 
the same course in the study of all phenomena '; it interprets them 
by the same language, as if to attest the unity and simplicity of 
the plan of the universe, and to make still more evident that 
unchangeable order which presides over aU natural causes. 
The problems of the theory of heat present so many examples 
of the simple and constant dispositions which spring from the 
general laws of nature; and if the order which is established in 
these phenomena could be grasped by our senses, it would produce 
in us an impression comparable to the sensation of musical sound. 
The forms of bodies are infinitely varied; the distribution of 
the heat which penetrates them seems to be arbitrary and confused ; 
but all the inequalities are rapidly cancelled and disappear as time 
passes on. The progress of the phenomenon becomes more regular 
and simpler, remains finally subject to a definite law which is the 
same in all cases, and which bears no sensible impress of the initial 
arrangement. 
All observation confirms these consequences. The analysis 
from which they are derived separates and expresses clearly, 1° the 
general conditions, that is to say those which spring from the 
natural properties of heat, 2° the effect, accidental but continued, 
of the form or state of the surfaces; 3° the effect, not permanent, 
of the primitive distribution. 
In this work we have demonstrated all the principles of the 
theory of heat, and solved all the fundamental problems. They 
could have been explained m~re concisely by omitting the simpler 
problems, and presenting in the first instance the most general 
results; but we wished to shew the actual origin of the theory and 
Digitized by Coogle 
PUELUrlIN.!RY DISCOURSE. 9 
its gradual progress. When this knowledge has been acquired 
and the principles thoroughly fixed, it is preferable to employ at 
once the most extended analytical methods, as we have done in 
the later investigations. This is also the course which we shall 
hereafter follow in the memoirs which will be added to this work, 
and which will form in some manner its complement 1; and by this 
means we shall have reconciled, so far as it can depend on our-
selves, the necessary development of principles with the precision 
which becomes the applications of analysis. 
The subjects of these memoirs will be, the theory of radiant 
heat, the problem of the terrestrial temperatures, that of the 
temperature of dwellings, the comparison of theoretic results with 
th~ which we have observed in different experiments, lastly the 
demonstrations of the differential equations of the movement of 
heat in fluids. 
The work which we now publish has been written a long time 
since; different circumstances have delayed and often interrupted 
the printing of it. In this interval, science has been enriched by 
important observations; the principles of our analysis, which had 
not at first been grasped, have become better known; the results 
which we had deduced from them have been discussed and con-
firmed. We ourselves have applied these principles to new 
problems, and have changed the form of some of the proofs. 
The delays of publication will have contributed to make the work 
clearer and more complete. 
The subject of our first analytical investigations on the transfer 
of heat was its distribution amongst separated masses; these have 
been preserved in Chapter Ill, Section II. The problems relative 
to continuous bodies, which form the theory rightly so calJed, were 
solved many years afterwards; this theory was explained for the 
first time in a. manuscript work forwarded to the Institute of 
Frunce at the end of the year 1807, an extract from which was 
published in the Bulletin des &ience8 (Societe Philomatique, year 
1808, page 112)~ We added to this memoir, and successively for-
warded very extensive notes, concerning the convergence of series, 
the diffusion of heat in an infinite prism, its emission in spaces 
1 These memoirs were never collectively published as a sequel or complement 
to the TIIeork .dnalytiqut dt fa Chaleur. But, DB will be seen presently, the author 
had written most of them before the pUblication of that work in 1822. [A. F.] . 
Digitized by Coogle 
10 THEORY OF BE!. T. 
void lIf air, the constructions suitable for exhibiting the chief 
theorems, and the analysis of the periodic movement at the sur-
face of the earth. Our second memoir, on the propagation of 
heat, was deposited in the archives of the Institute, on the 28th of 
September, 1811. It was formed out of the preceding memoir and 
the notes already· sent in; the geometrical constructions and 
those details of analysis which had no neceB88.I'Y relation to the 
physical problem were omitted, and to it was added the general 
equation which expreBBeB the state of the surface. This second 
work was sent to press in the course of 1821, to be inserted in 
the collection of the Academy of Sciences. It is printed without 
any change or addition; the text agrees literally with the deposited 
manuscript, which forms part of the archives of the Institute 1. 
In this memoir, and in the writings which preceded it, will be 
found a fil'Bt explanation of applications which our actual work 
.1 It appears as a memoir and supplement in volumes IV. and V. of the Mi· 
molru th Z' .lfcadlmie del 8cimeu. For cODvenience of comparison with the table 
of contenw of the .lfMl"ticlll TMory 0/ Heat, we subjoin the titles and heads of 
the chapters of the printed memoir : 
TJdoBlB DU KOUTBKBN'r DB Ll OJIALJIUB D.-a LBa COBPI BOLIDBI, P.lB M. 
FoUBlBB. [Mblloiru th f.lfcadlmie Ror/ale del SClenCU th Z'IMtit'" th Frame. 
TollUl IV. (for year 1819). Parill8t'-l 
I. Ezporitiora. 
n. NotioM ghllralu et cUjlllitioM pr6Umina.lru. 
m. EquatioM tlu moIWemetlt de l4 chaleur. 
IV. Du mouvern.emJiniaireet "arU de l4 chaleur daM une a"""Ue. 
V. De l4 propagatiora th l4 chaleur daM une l4me rectafllUlairedOllt lu tetnplraturu 
IOIIt coutalltU. 
VI. De l4 co""""nicatioll th la. chaleur entre del IIIIJItUdWjoilltU. 
vn. Du _"emetlt "am de l4 chaleur daM VRe 'PUre .0Uth. 
VIII. Du _"emme "arilf de l4 chaleur daM Ufl Cflliftdre .olith. 
IX. De la. propa.gatioll th l4 chaleur daM UII priIme do'" Z'eztl'6111iU "" CllltfitUie 
a URe temp6ra.ture _taRte. 
X. Du _"_fit "am th l4 chaleur da", UII .olith th /Orftlll cubique. 
XL Du _"ern.em ZiMaire et "am de l4 chaleur da", let cOrpl dofIt URe dimenriora 
ut inftnie. 
BUlD DU '-OIU mTrrULB: TDoBlB DU KOUTBJIBN'r DB Ll OJIALJIUB D.-I 
LBa COBPa SOLlDBB j P.lB M. FouBIBB. [Mimolru th Z' .lfcacUmiil Rot/ale del SciilRC/I, 
de fI",titut de France. Tome V. (for year 1820). Paril,1826.] 
m. Du tempbature, terrutru, /It du _"emetlt de l4 chaleur cia", Vint6ritUr 
d't.CIIt 'PUre roUth, dofIt la.lIlf'/aclI elt a.rlUjdtie a del changtmeM pbiodique, 
th t/lmp6rature. 
xm. Du loi. _tM_tiqvet th flquilibre th l4 chaleur rayoraMftte. 
XlV. Compa.railora del r61ultatl de l4 ,Marie a,,1IC CtUZ th divtTlIII ezp6rimee •• 
(A. F.] 
Digitized by Coogle 
PBELDDNARY nmCOUBS& 11 
does not contain; they will be treated in the subsequent memoirs 1 
at greater length, and, if it be in our power, with greater clear-
ness. The resnIts of our labours concerning the same problems 
are also indicated in several articles already published. The 
extract inserted in the Annalea de Ohimie et de Physique shews 
the aggregate of our researches (Vol. III. page 350, year 1816). 
We published in the Annalu two sepamte notes, concerning 
radiant heat (VoL IV. page 128, year 1817, and Vol. VI. page 259, 
year 1817). 
Several other articles of the same collection present the most 
constant results of theory and observation; the utility and the 
extent of thermological knowledge could not be better appreciated 
than by the celebrated editors ofthe Annales·. 
In the Bulletin des Sciences (Societe philomatique year 1818, 
page 1, and year 1820, page 60) will be found an extract from 
a memoir on the constant or variable temperature of dwellings, 
and an explanation of the chief consequences of our analysis of 
the terrestrial temperatures. 
M. Alexandre de Humboldt, whose researches embrace all the 
great problems of natural philosophy, has considered the obser-
vations of the temperatures proper to the different climates 
from a novel and very important point of view (Memoir on Iso-
thermal lines,8ociiU d'Arcueil, VoL w. page 462); (Memoir on 
the inferior limit of perpetual snow, Annales de Ohimi6 Bt de 
Physique, VoL v. page 102, year 1817). 
As to the differential equations of the movement of heat in 
fluids' mention bas been made of them in.the annual history of 
the Academy of Sciences. The extract from our memoir shews 
clearly its object and principle. (Analyse des trava~ de f Aea-
aemie des Sciences, by M. De Lambre, year 1820.) 
The examination of the repUlsive forces produced by beat, 
which determine the statical properties of gases, does not belong 
1 See note, page 9, and the notes, pages 11-18. 
I Gay.LUS88C and Arago. Bee Dote, p. 18 • 
• Mlmoira tU r .dcadlmil da Scimcu, TOfIW XII., Pam, 1888, contain on pp. 
607-61" Mlmoire tl/JftQJ,.. _l. _"emenC tU lei chaleur damlu jluitU., par M. 
Fourier. Lv a ".dca4l1nil Borak ,u, SciIftcu, 4 &po 1820. It is followed on pp. 
615-680 by Eztrait lit. fIOU. _HUlm'u comertllt. par Z'au'tur. The memoir 
is signed Jh. Fourier, Paris, 1 Bep. 1820, but was published after the death of the 
author. [A. P.) 
Digitized by Coogle 
12 THEORY OF HEAT. 
to the analytical subject which we have colUlidered. This question 
connected with the theory of radiant heat has just 1>een discussed 
by the illustrious author of the Yecanique celeste, to whom aU 
the chief branches of mathematical analysis owe important 
discoveries. (Connai8sance des Temps, years 1824-5.) 
The new theories explained in our work are united for ever 
to the mathematical sciences, and rest like them on invariable 
foundations; an the elements which they at present possess they 
will preserve, and will continually acquire greater extent. Instru-
ments will be perfected and experiments multiplied. The analysis 
which we have formed will be deduced from more general, that 
is to say, more simple and more fertile methods common to many 
classes of phenomena. For all substances, solid or liquid, for 
vapours and permanent gases, determinations will be made of all 
the specific qualities relating to heat, and of the variations of the 
coefficients whioh express them t. At different stations on the 
earth observations will be made, of the temperatureR of the 
ground at different depths, of the intensity of the solar heat and 
its effects, constant or variable, in the atmosphere, in the ocean 
and in lakes; and the constant temperature of the heavens proper 
to the planetary regions will become known I. The theory itself 
1 M4moirt. tk l' .dcad4mu dt. ScUncu, T07M VIII., Pari. 1829, contain on 
pp. 581-622, Mimoire IVr la TMorie .dnalytiqllt dt la Chaleur, par M. Fourier. 
This was published whilst the author was Perpetual Secretary to the Academy. 
The first only of four parts of the memoir is printed. The contents of all are 
stated. I. Determines the temperature at any point of a prism whose terminal 
temperatures are funotions of the time, the initial temperature at any point being 
a funotion of it. distance from one end. fi. Examines the ohief consequences of 
the general solution, and applies it to two distinct oases, according as the tempe-
ratures of the ends of the heated prism are periodio or not. m. Is historioal, 
&numerates the earlier experimental and anaIytioal researches of other Writers 
relative to the theory of heat; considers the nature of the transoendental equations 
appearing "in the theory; remarks on the employment of arbitrary funotions: 
replies to the objections of M. Poisson; adds some rentarks on a problem of the 
motion of waves. IV. Extends the application of the theory of heat by taking 
account, in the analysis, of variations in the speciflo coefficients which measure 
the capacity of substances for heat, the permeability of solids, and the penetra-
l1ility of their ·smaoes. [A. F.] 
1 M4moirt. tk l'.dcaMmie du ScUftce., Tome VII., Pam, 1827, oontain on 
pp. 569-604, M4moire IVr lei kmpiraturu du globe krrutrt lit du eqactl planl-
Caire., par M. Fourier. The memoir is entirely desorlptive; it was read before the 
Academy, 20 and 29 Bop. 1824 (Annalu de Chimie d tk PhyrilJ'U, 1824, XXVII. 
p.136). [A. F.] 
Digitized by Coogle 
PRELIHINARY DlSCOURSE. 13 
will direct aU these measures, and assign their precISIOn. No 
considerable progress can hereafter be made which is not founded 
on experiments such as these j for mathematical analysis can 
deduce from general and simple phenomena the expression of the 
laws of nature; but the special application of these laws to very 
complex effects demands a long series of exact observations. 
The complete list of the Articles on Heat, published by H. Fourier, in the 
A.MlJlu de Chimie el de Phylique, Smu 2, is as follows : 
1816. m. pp. 850-875. Thlurie de la Chaleur (Eztrait). Description by the 
author of the 4to volume afterwards published in 1822 without the chapters on 
radiant heat, solar heat as it affects the earth, the comparison of analysis with 
experiment, and the history of the rise and progress of the theory of heat. 
1817. IV. pp. 12~145. Note IUr la Chaleur rayonnante. Mathematical 
.ketch on the sine law of emiBBion of heat from a surface. Provea the author's 
pal'lAlox on the hypotheais of equal intensity of emission in all directions. 
1817. VL pp. 259-803. Que,tiona IUr la tMorie phylique de la chaleur 
rayomaanfll. An elegant physieal treatise on the discoveriea of Newton, Pictet, 
Wells, Wollaston, Leslie and Prevosi. 
1820. XIII. pp. 418--438. Sur,le re!roidil,ment ,~culatre de la fIlrre (Eztrail). 
Sketch ofa memoir, mathematical and descriptive, on the waste of the earth's 
initial heal 
1824. XXYn. pp. 136-167. Remarque, ghbale, fUr let temp4rature, du glob, 
fIlrrutre et de, elp4Cu plamtaire,. This is the deacriptive memoir referred to 
above, MhR. A.cad. d. Sc. Tome VII. 
1824. XXVIL pp. 236-281. R&umI thlorique de, propri~U. de la chaleur 
raYOfttl4nfll. Elementary analytical account of surface-emiBBion and absorption 
baaed on the prinoiple of equilibrium of temperature. 
1825. xxvm. pp. 887-866. Remarque. IUr la thlorW mathlmatique de la 
chaleur rayonnante. Elementary analysis of emission, absorption and rellection 
by walls of enclosure uniformly heated. At p. 364, H. Fourier promises a TMurie 
phyrique de la chaleur to contain the applications of the ThlorW A.nalytique 
omilied from the work published in 1822. 
1828. XXXVIL pp. 291-816. Recherchu uplrifllllfttale. "'" la !aculU con-
duetrice de. carpi minct •• oumil a l'action de la chaleur, et de.cription d'un nouveau 
'hmRom~e de contact. A thermoscope of contact intended for lecture demonstra. 
tions iB alBo described. H. Emile Verdet in his Con!~rmcu de Phy,ique, Paril, 
1872. Part L p. 22, has stated the practical reasons against relying on the 
theoretical indications of the thermometer of contact. [A. F.] 
Of the three no.tieea of memoirB by H. Fourier, contained in tbe Bulletin de, 
Sciencu par la Sacilll Philomatique, and quoted here at pages 9 and 11, the first 
was written by H. Poisson, the mathematical editor of the Bulletin, the other two by 
H. Fourier. [A. F.) 
Digitized by Coogle 
THEORY OF HEAT. 
CHAPTER I. 
INTRODUCTION. 
FIRST SECTION. 
8tatemM&t of eM Object of eM Work. 
1. THE effects of heat are subj~t to constant laws which 
cannot be discovered without the aid of mathematical analysis. 
The object of the theory which we are about to explain is to 
demonstrate these laws; it reduces all physical researches on 
the propagation of heat, to problems of the integral calculus 
whose elements are given by experiment. No subject has more 
extensive relations with the progress of industry and the natural 
sciences; for the action of heat is always present, it penetrates 
all bodies and spaces, it influences the processes of the arts, 
and occurs in all the phenomena of the universe. 
When heat is unequally distributed among the different parts 
of a solid mass, it tends to attain equilibrium, and passes slowly 
from the parts which are more heated to those which are less; 
and at the same time it is dissipated at the surface, and lost 
in the medium or in the void. The tendency to uniform dis. 
tribution and the spontaneous emission which acts at the surface 
of bodies, change continually the temperature a.t their different 
points. The, problem of the propagation of heat consists in 
1 Cf. Plato, Tinuew, 68, B. 
1ST. /I' brex'lPfiT'O /CotIp.EIIIStu. rei "P, np "pWrOP /Cal oyijeo n1 Up« n1 (;1Iwp .... , .... 
'''''X'IJ'f&rClla.ro [elBfOr) dlfll' re /Cal dpcS"""f. [Ao F.) 
Digitized by Coogle 
cu. L SECT. I.] INTRODUCTION. 15 
determining what is the temperature at each point of a body 
at a given instant, supposing that the initial temperatures are 
known. The following examples will more clearly make known 
the nature of these problems. 
2. If we expose to the continued and uniform action of a 
source of heat, the same part of a meta1lic ring, whose diameter 
is large, the molecules nearest to the source will be first heated, 
and, after a certain time, every point of the solid will have 
acqUired very nearly the highest temperature which it can attain. 
This limit or greatest temperature is not the same at different 
points; it becomes less and lesll according as they become more 
distant from that point at which the source of heat is directly 
applied. 
When the temperatures have become permanent, the source 
of heat supplies, at each instant, a quantity of heat which exactly 
compensates for that which is dissipated at all the points of the 
external surface of the ring. 
If now the source be suppressed, heat will continue to be 
propagated in the interior of the solid, but that which is lost 
in the medium or the void, will no longer be compensated as 
formerly by the supply from the source, so that all the tempe-
ratures will vary and diminish incessantly until they have be-
come equal to the temperatures of the surrounding medium. 
3. Whilst the temperatures are permanent and the source 
remains, if at every point of the mean circumference of the ring 
an ordinate be raised perpendicular to the plane of the ring, 
whose length is proportional to the fixed temperature at that 
point, the curved line which passes through the ends of these 
ordinates will represent -the permanent state of the temperatures, 
and it is very easy to determine by analysis the nature of this 
line. It is to be remarked that the thickness of the ring is 
supposed to be sufficiently small for the temperature to be 
sensibly equal at all points of the same section perpendicular 
to the mean circumference. When the source is removed, the 
line which bounds the ordinates proportional to the temperatures 
at the different points will change its form continually. The 
problem consists in expressing, by one equation, the variable 
Digitized by Coogle 
16 THEORY OF HEAT. [CHAP. I. 
form of this curve, and in thus including in a single formula 
all the successive states of the solid. 
4. Let. be the constant temperature at a point m of the 
mean circumference, te the distance of this point from the source, 
that is to say the length of the arc of the mean circumference, 
included between the point m and the point 0 which corresponds 
to the position of the source; z is the highest temperature 
which the point m can attain by virtue of the constant action 
of the source, and this permanent temperature z is·a function 
fete) of the distance te. The first part of the problem consists 
in determining the function j(te) which represents the permanent 
state of the solid. 
Consider next the variable state which succeeds to the former 
state as soon as the source has been removed; denote by t the 
time which has passed since the suppression of the source, and 
by 11 the value of the temperature at the point m after the 
time t. The quantity tJ will be a certain function F (a:, t) of 
the distance te and the time t; the object of the problem is to 
discover this function F (te, t), of which we only know as yet 
that the initial value is f (0:), so that we ought to have the 
equation J (:r) = F (0:, 0). 
5. If we place a solid homogeneous mass, having the form 
of a sphere or cube, in a medium maintained at a constant tem-
perature, and if it remains immersed. for a very long time, it will 
acquire at all its points a temperature differing very little from 
that of the fluid. Suppose the mass to be withdra\Vll in order 
to transfer it to a cooler medium, heat will begin to be dissi. 
pated at its surface; the temperatures at different points of the 
mass will not be sensibly the same, and if we suppose it divided 
into an infinity of layers by surfaces parallel to its external sur-
face, each of those layers will transmit, at each instant, a certain 
quantity of heat to the layer which surrounds it. If it be 
imagined that each molecule carries a separate thermometer, 
which indicates its temperature at every instant, the state of 
the solid will from time to time be represented by the variable 
system of all these thermometric heights. It is required to 
express the successive. states by analytical formulre, so that we 
Digitized by Coogle 
SECT. I.] INTRODUCTION. 17 
may know at any given instant the temperatures indicated by 
each thermometer, and compare the quantities of heat which 
flow during the same instant, between two adjacent layers, or 
into