374 THEORY OF HEAT. [CHAP. IX. 



and putting for u its value =&amp;gt; we have 



2 V t 



_ 2 



e *t ,~ 



dn ff~*** cos nx = VTT. 



pt 



Moreover the particular value j=- is simple enough to present 



itself directly without its being necessary to deduce it from the 

 value e~ nH cosnx. However it may be, it is certain that the 



-& dv d*v 



function j=- satisfies the differential equation -j- = -^ it is the 



(j?~q) 



6~~ ^t 



same consequently with the function ^ , whatever the quan- 



*Jt 



tity a may be. 



376. To pass to the case of three dimensions, it is sufficient 



_&M? 



to multiply the function of x and t, ^ , by two other similar 



ijt 



functions, one of y and t, the other of z and t\ the product will 

 evidently satisfy the equation 



dv _ d*v d?v d?v 

 dt~d^ + dy z + d? 



We shall take then for v the value thus expressed : 



If now we multiply the second member by den, d$, dy, and by 

 any function whatever/ (a, /3, 7) of the quantities a, /6, 7, we find, 

 on indicating the integration, a value of v formed of the sum of an 

 infinity of particular values multiplied by arbitrary constants. 



It follows from this that the function v may be thus ex 

 pressed : 



M-oo ,.+00 -+00 ^3 (q-^)2 + (.8-y) 2 +(Y-g) 2 



J-oo J-oo J -OP 



This equation contains the general integral of the proposed 

 equation (A): the process which has led us to this integral oug^t^ 



