SECT. II.] THE CASE OF THREE DIMENSIONS. 371 



374. Since the function v or c/&amp;gt; (x, y, z, t] represents the 

 initial state when in it we make t = 0, and since it satisfies the 

 differential equation of the propagation of heat, it represents also 

 that state of the solid which exists at the commencement of the 

 second instant, and making the second state vary, we conclude 

 that the same function represents the third state of the solid, and 

 all the subsequent states. Thus the value of v, which we have 

 just determined, containing an entirely arbitrary function of three 

 variables x, y, z, gives the solution of the problem ; and we cannot 

 suppose that there is a more general expression, although other 

 wise the same integral may be put under very different forms. 



Instead of employing the equation 



we might give another form to the integral of the equation 



-77 = -j-g ; and it would always be easy to deduce from it the 

 ctt dx 



integral which belongs to the case of three dimensions. The 

 result which we should obtain would necessarily be the same as 

 the preceding. 



To give an example of this investigation we shall make use of 

 the particular value which has aided us in forming the exponential 

 integral. 



Taking then the equation -^- = ^-j ... (b), let us give to v the 

 very simple value e~ nH cosnx, which evidently satisfies the 

 differential equation (6). In fact, we derive from it -j- = rfv 



d*v 

 and -y-g = ri*v. Hence also, the integral 



CUD 



r 



V m 



dn e~ nZt cosnx 



belongs to the equation (6) ; for this value of v is formed of the 

 sum of an infinity of particular values. Now, the integral 







nx 



242 



