250 THEORY OF HEAT. [CHAP. IV. 



(n 1); now it is easy to prove that if we give to j its n successive 

 values, from to (n 1), the sum 



2 cos (jj, v) q 4- cos 1 (fL v) q + ~ cos 2 (p v) q 



+ = cos 3 (fjL v) q + . . . + ~ cos (n - 1) (p - v) q 



A Z 



has a nul value, and that the same is the case with the series 

 ^ cos (JM + v) q + cos 1 (p + v) q + ^ cos 2 (p + v) q 



+ 2 cos 3 (/A + v) ^ + . . . + g cos ( n ~ 1) (^ + &quot;) 

 In fact, representing the arc (p v)q by or, which is consequently 



2-7T 



a multiple of , we have the recurring series 



cos Oa, cos 1#, cos 2z, . . . cos (w 1) a, 

 whose sum is nothing. 



To shew this, we represent the sum by s, and the two terms of 

 the scale of relation being 2 cos a and 1, we multiply successively 

 the two members of the equation 



s = cos Oa + cos 2a + cos 3a + . . . + cos (n 1) a 



by 2 cos a and by + 1 ; then on adding the three equations we 

 find that the intermediate terms cancel after the manner of re 

 curring series. 



If we now remark that not. being a multiple of the whole cir 

 cumference, the quantities cos (n 1.) a, cos (n 2) a, cos (n 3) a, 

 &c. are respectively the same as those which have been denoted 

 by cos ( a), cos ( 2a), cos ( 3a), ... &c. we conclude that 



2s 25 cos a = ; 



thus the sum sought must in general be nothing. In the same 

 way we find that the sum of the terms due to the development of 

 \ cos j (IJL -f v) q is nothing. The case in which the arc represented 

 by a is must be excepted ; we then have 1 - cos a = 0; that is 

 to say, the arcs it and v are the same. In this case the term 

 J cos,/ (jj, + v) q still gives a development whose sum is nothing ; 



