DEMONSTRATION OF THE COTESIAN THEOREM. 281 



Now these equations must necessarily have one common Demonstration 

 root, being both obtained from the same value of y\ and ^ e C ° e te f S 

 farther, since both equations are reciprocal ones, if y be 



one root, — is another root, and the first equation having? 



y 



but two roots, and such of those being also roots of the se- 

 cond equation, it follows that the former of our formulae is 

 a divisor of the latter. And this is true, if we change y 

 into q y, without, however, altering the value of 2 cos. x, 

 and 2 cos. mx; for by substituting for these their true value 



y -\ and y m -\ , and q y for y, the above formulae are 



reduced to 



1st, fay* — l) X (9 — 1) 



2d, (fy sm -l) X (q m — 1) 



The former of which is evidently a divisor of the latter, en- 

 tirely independent of the value of q, or of the measure of 

 the angle represented by x. 



We may therefore, instead of x, write , then 



m 



our formulae will become 



i z ^ smx -\- nc\ 

 q*y*~2qy cos. (^— — J + l 



5 !m y sm — 2 g m ^ m cos. (in x + n c) + 1 

 The former being still a divisor of the latter. 



We may here also observe, that while c represents the en- 

 tire circumference, the cos. (m x -+■ w f) — cos. m x, what- 

 ever integral value we give to n; and hence making re- 

 spectively n = 0, 1, 2, 3, &c, m — 1, it follows, that the 

 formula 



qim yim cj, qra ytn cos (m x -{- n c) + 1 , 



has for its divisors the m following formulae, 



9 V — 2?ycos. ^— — J -f i 



q y —iqy COS. (^ — J -f i 



a y — 2 q y cos. (^ J + 1 



