SECT. IV.] SINGLE LAYER INITIALLY HEATED. 449 



particular cases excepted, we may say that a function subject to a 

 continuous law, which forms the first member of equations of this 

 kind, does not coincide with the function expressed by the second 

 member, except for values of x included between and X. 



Properly speaking, equation (e) is an identity, which exists 

 for all values which may be assigned to the variable x\ each 

 member of this equation representing a certain analytical function 

 which coincides with a known function f(x) if we give to the 

 variable x values included between and A 7 &quot;. With respect to the 

 existence of functions, w T hich coincide for all values of the variable 

 included between certain limits and differ for other values, it is 

 proved by all that precedes, and considerations of this kind are a 

 necessary element of the theory of partial differential equations. 



Moreover, it is evident that equations (e) and (E) apply not 

 only to the solid sphere whose radius is X, but represent, one the 

 initial state, the other the variable state of an infinitely extended 

 solid, of which the spherical body forms part ; and when in these 

 equations we give to the variable x values greater than X, 

 they refer to the parts of the infinite solid which envelops the 

 sphere. 



This remark applies also to all dynamical problems which are 

 solved by means of partial differential equations. 



427. To apply the solution given by equation (E) to the case 

 in which a single spherical layer has been originally heated, all 

 the other layers having nul initial temperature, it is sufficient to 



take the integral \dj. sin (/^a) aF (a) between two very near limits, 



a = r, and a = r + u, r being the radius of the inner surface of the 

 heated layer, and u the thickness of this layer. 



We can also consider separately the resulting effect of the 

 initial heating of another layer included between the limits r + u 

 and r + 2u ; and if we add the variable temperature due to this 

 second cause, to the temperature which we found when the first 

 layer alone was heated, the sum of the two temperatures is that 

 which would arise, if the two layers were heated at the same time. 

 In order to take account of the two joint causes, it is sufficient to 

 F. H. 29 



