450 THEORY OF HEAT. [CHAP. IX. 



take the integral Ida sin (/i 4 ot) aF(a) between the limits a r and 



a = r + 2w. More generally, equation (E) being capable of being 

 put under the form 



f x j -vi \ 

 v = I ay. . ctr (a) sin /^a 







sm W e 



x \ d/3 si 

 Jo 



sn uj sn 



we see that the whole effect of the heating of different layers is 

 the sum of the partial effects, which would be determined separately, 

 by supposing each of the layers to have been alone heated. The 

 same consequence extends to all other problems of the theory of 

 heat ; it is derived from the very nature of equations, and the form 

 of the integrals makes it evident. We see that the heat con 

 tained in each element of a solid body produces its distinct effect, 

 as if that element had alone been heated, all the others having 

 nul initial temperature. These separate states are in a manner 

 superposed, and unite to form the general system of temperatures. 



For this reason the form of the function which represents the 

 initial state must be regarded as entirely arbitrary. The definite 

 integral which enters into the expression of the variable tempera 

 ture, having the same limits as the heated solid, shows expressly 

 that we unite all the partial effects due to the initial heating of 

 each element. 



428. Here we shall terminate this section, which is devoted 

 almost entirely to analysis. The integrals which we have obtained 

 are not only general expressions which satisfy the differential equa 

 tions ; they represent in the most distinct manner the natural effect 

 which is the object of the problem. This is the chief condition which 

 we have always had in view, and without which the results of in 

 vestigation would appear to us to be only useless transformations. 

 When this condition is fulfilled, the integral is, properly speaking, 

 the equation of the phenomenon; it expresses clearly the character 

 and progress of it, in the same manner as the finite equation of a 

 line or curved surface makes known all the properties of those 

 forms. To exhibit the solutions, we do not consider one form only 

 of the integral ; we seek to obtain directly that which is suitable 

 to the problem. Thus it is that the integral which expresses the 



