CHAP. VIII.] GENERAL VALUE OF V. 325 



The second member is formed of the product of the three 

 factors written in the three horizontal lines, and the quantities 

 a x , a 2 , 3 , &c. are unknown coefficients. Now, according to the 

 hypothesis, if t be made = 0, the temperature must be the same at 

 all points of the cube. We must therefore determine a 1} a 2 , a 3 , &c., 

 so that the value of v may be constant, whatever be the values of 

 x, y, and z, provided that each of these values is included between 

 a and a. Denoting by 1 the initial temperature at all points of 

 the solid, we shall write down the equations (Art. 323) 



1 = a : cos n^x + a 2 cos n z x + a a cos n s x + &c., 

 1 = & x cos n t y + 6 a cos n 2 y + b 3 cos n^y + &c., 

 1 = c l cos n^z + c a cos n z z + c a cos n B z + &c., 



in which it is required to determine a lt a t , a s , &c. After multi 

 plying each member of the first equation by cosnx, integrate 

 from # = to X CL-. it follows then from the analysis formerly 

 employed (Art. 324) that we have the equation 



sin n^a cos n^x sin n^a cos n^x sin n z a cos njc 



1 = i : ^T? r -f i : ^-s r + , : gin 



tn^\ 

 nja, ) 



+ &c. 

 Denoting by ^ the quantity ^ f 1 H * j, we have 



_ . sin n.a sin n.a sin n.a p 



1 = cos njc -\ cos n^x H ^ cos n s x -f &c. 



This equation holds always when we give to x a value included 

 between a and a, 



From it we conclude the general value of v, which is given by 

 the following equation 



/sin n. a ,. 2/ sin n a , ,. \ 



v = ( L cos n^x e~ kni t -f cos njc e~ kn * f + &c. ) , 



( s - i- cos njje~ kniH ^ cos n$ e~ ina ^ + &c.J, 



/sin n CL , ,, sin n n a 



ros M z fi ~ kn * f -I si cos n^z e 



