SECT. II. 



VALUES OF THE COEFFICIENTS. 



If now in the values of a, b, c, d, &c., we notice what are the 

 factors which must be joined on to numerators and denominators 

 to complete the double series of odd and even numbers, we find 

 that the factors to be supplied are : 



-=V- , y whence we conclude . 



177. Thus the eliminations have been completely effected, 

 and the coefficients a, b } c, d, &c., determined in the equation 



1 = a cos y + b cos 3?/ + c cos 5y + d cos 7y + e cos 9# + &c. 



The substitution of these coefficients gives the following equa 

 tion : 



7T 



1 



- = COS 7/ - COS 



1 c 1 K 1 



-f ^COS 5?/ ^COS /^/+7^ COS 



o / 9 



- &c. 



The second member is a function of y, which does not change 

 in value when we give to the variable y a value included between 

 ^TT and -f |TT. It would be easy to prove that this series is 

 always convergent, that is to say that writing instead of y any 

 number whatever, and following the calculation of the coefficients, 

 we approach more and more to a fixed value, so that the difference 

 of this value from the sum of the calculated terms becomes less 

 than any assignable magnitude. Without stopping for a proof, 



1 It is a little better to deduce the value of & in or, of c in &, &c. [E. L. E.] 



2 The coefficients a, b, c, &c., might be determined, according to the methods 

 of Section vi. , by multiplying both sides of the first equation by cos y, cos 3?/, 



cos 5v, &c., respectively, and integrating from --Trto +^TT, as was done by 



& & 



D. F. Gregory, Cambridge Mathematical Journal, Vol. i. p. 106. [A. F.] 



