314 THEORY OF HEAT. [CHAP. VII. 



Thus the foregoing integral, which reduces to 



-2 - 2 ( n sm n l cos vlv cos nl sin vl), 



is nothing, except only in the case where n v. Taking then the 

 integral 



a jsin (n v)l sin (n + v) I] 



2 [ n-v n + v J 



we see that if we have n = v, it is equal to the quantity 



sin 2 



It follows from this that if in the equation 



1 = a i cos 71$ + 2 cos n 2 y + a s cos n z y + &c. 



we wish to determine the coefficient of a term of the second 

 member denoted by a cos ny y we must multiply the two members 

 by cos ny dy, and integrate from y = to y L We have the 

 resulting equation 



f l * * A sin2nZ\ 1 . 



cos nydy = -^a\l H - 1 = - sin nl, 

 Jo y J 2 V 2 / fi 



whence we deduce x ^ - . _ 7 = - a. In this manner the coeffi- 

 2nl + sin 2nl 4 



cients a^ a 2 , a 3 , &c. may be determined ; the same is the case 

 with b lt 6 2 , 6 3 , &c., which are respectively the same as the former 



coefficients. 



325. It is easy now to form the general value of v. 1st, it 



d?v d zv d?v 



satisfies the equation -Y-.+ T-^ + -T^ = O; 2nd. it satisfies the two 

 dx dy dz 



conditions k-j- + hv = 0, and Jc-j- + hv 0; 3rd, it gives a constant 



value to v when we make x 0, whatever else the values of y and 

 z may be, included between and Z; hence it is the complete 

 solution of the proposed problem. 



We have thus arrived at the equation 



cos n^y sin nj, cos n z y sin n s l cos n z y 

 in 2 2n7 + sin 2 2w^ + sin2w C J 



1 _ sin n cos n 

 ~ 



