252 THEORY OF HEAT. [CHAP. IV. 



or indeed 



cos Ou, cos Iw, cos 2w, cos 3u, ... cos (n 1) u, 



form a recurring periodic series composed of n terms : if we com- 



27T 

 pare one of the two series corresponding to an arc u or p. 



n 



with a series corresponding to another arc v or v , and 



multiply term by term the two compared series, the sum of the 

 products will be nothing when the arcs u and v are different. If 

 the arcs u and v are equal, the sum of the products is equal to |-/?, 

 when we combine two series of sines, or when we combine two 

 series of cosines ; but the sum is nothing if we combine a series of 

 sines with a series of cosines. If we suppose the arcs u and v to 

 be nul, it is evident that the sum of the products term by term is 

 nothing whenever one of the two series is formed of sines, or when 

 both are so formed, but the sum of the products is n if the com 

 bined series both consist of cosines. In general, the sum of the 

 products term by term is equal to 0, or \n or n ; known formulae 

 would, moreover, lead directly to the same results. They are pro 

 duced here as evident consequences of elementary theorems in 

 trigonometry. 



271. By means of these remarks it is easy to effect the elimi 

 nation of the unknowns in the preceding equations. The unknown 

 A v disappears of itself through having nul coefficients ; to find B^ 

 we must multiply the two members of each equation by the co 

 efficient of B t in that equation, and on adding all the equations 

 thus multiplied, we find 



To determine A 2 we must multiply the two members of each 

 equation by the coefficient of A 9 in that equation, and denoting 



the arc - - by q, we have, after adding the equations together, 



W9 



a l sin 0^ 4- a 2 sin Iq + a s sin 2q + . . . -f a n sin (n l)q = 

 Similarly to determine B a we have 

 rtj cos 0^ 4- a z cos 1 q + a a cos 2&amp;lt;/ f . . . + a n cos (n - 1) q = ^ n 



