SECT. I.] COEFFICIENTS OF THE SOLUTION. 275 



order, beginning with the least, we form the general equa 

 tion 



vx = a~ ltn ? i sin njc + a 2 e~ kn & sin w 2 # + a 3 e~ kna2t sin n s x + &c. 



If t be made equal to 0, we have as the expression of the 

 initial state of temperatures 



vx = a x sin n t x + a z sin n 2 x + a z sin n 3 x -f &c. 



The problem consists in determining the coefficients a lt a 2 , a 3 

 &c., whatever be the initial state. Suppose then that we know 

 the values of v from x = to x = X, and represent this system of 

 values by F(x) ; we have 



F(x) = - (a x sin n^x + 2 sin njc + a s sin n s x + a 4 sin n^x + &C.) 1 . . . (e). 



2.91. To determine the coefficient a lt multiply both members 

 of the equation by x sin nx dx, and integrate from x = to x = X. 



The integral Ismmx sin nx dx taken between these limits is 



5 2 ( m sin nXcos mX+ n sin mJTcos wX). 



m n 



If m and w are numbers chosen from the roots w 1 , w 2&amp;gt; w 3 , 



&c., which satisfy the equation - ^= 1 hX, we have 



tan TL^\. 



mX nX 



tanmX t& 



or m cos m X sin w X n sin w X cos w JT = 0. 



We see by this that the whole value of the integral is nothing; 

 but a single case exists in which the integral does not vanish, 



namely, when m = n. It then becomes ^ ; and, by application of 

 known rules, is reduced to 



-- 

 2 4sn 



1 Of the possibility of representing an arbitrary function by a series of this 

 form a demonstration has been given by Sir W. Thomson, Camb. Math. Journal, 

 Vol. m. pp. 2527. [A, F.] 



182 



