FIFTH REPORT- 1835. 
24 
But without further examining this point, or the general 
question of the reality of the molecular hypothesis, which de¬ 
pends mainly on the same considerations, I shall proceed to the 
next division of the subject. 
3. Mathematical Solutions of the Equations. —In the same 
memoirs of Fourier in which he gave the differential equations 
for the motion of heat in various cases, he also gave the integrals 
of these equations in some of the most important instances. 
These solutions were obtained by means of very peculiar artifices 
and have led to, or been connected with, some remarkable dis¬ 
quisitions on points of pure analysis. It does not belong to my 
purpose to give any account of the labours of writers on heat 
in this point of view; but in order to bring under the reader’s 
notice some of the leading features of the inquiry, I will briefly 
refer to two of the problems, the motion of heat in a rectangular 
plate, and in a solid sphere. 
The first of the special problems treated by Fourier* is to find 
the ultimate and permanent distribution of heat in a rectangular 
lamina, of which one side is kept uniformly hot, the two adja¬ 
cent ones uniformly cold, and the fourth side is at an indefinite 
distance. The equation belonging to this case is of a simple 
and well-known formf; and it is easy to obtain a possible so¬ 
lution of itj, which exhibits sufficiently the general course of 
the phenomena; namely, that in proceeding along the lamina 
from the hot end, the temperature diminishes in geometrical 
progression at equal distances ; but that in proceeding across 
the lamina from the middle to each cold side, the temperature 
diminishes according to the law of a cosine. This possible case 
is, however, merely a particular solution; and in order to make 
the solution complete, we must be able to extend it so that the 
temperature of the boundaries of the lamina shall be regulated 
by any prescribed law; for example, so that at the hot end it 
shall be uniform from one side to the other, instead of dimi¬ 
nishing from the middle each way, as it would in the case just 
stated. And this consideration introduces us to a very remark¬ 
able province of analysis ; for it is easy, by adding together any 
number of such particular solutions as we have mentioned, to 
produce a function which gives a more general solution § ; but 
in order to satisfy the condition just stated, this function must 
* Theorie de la Chaleur. 
d 2y ^2 v 
f + ——, = ; 0 where x is parallel to the cold sides, y to the hot one. 
dx* ay* 1 
J — ms 
X v = g cos m y. 
— ms . , —m's i • n 
§ v = a g cos my a i cos m y -+- a s 
— m "s 
cos in' y &c. 
