and on Foiirier*s Tlieorem. 493 



deduced therefrom, will be identical with those given by 

 the formulae (A.). But if y = 0, we have, when j/ = go 



(observing that since q = (ifQO'i 9 = 0, and fl = — — - j, 

 fyv oosr>,= sin ri,{^ + £^' + £l(£^|)l!l!+ &c. 

 — sni ry . -— 



= -^sin 00 (oo)^ 



= 00 , if ^ be finite. 

 But if ^ be zero (as it may be), we have 



/ 



cos 7y = -yr sin co ( oo )o = — - sin oo. 



In the same manner we should find that when y is in- 

 finite, the above expression for the indefinite integral 



/ sin ly = ^ cos oo. Hence it is not true that the for- 

 mulae of Euler give definite values for sin oo and cosoc ; the 

 fact being that they leave those functions just as they found 

 them, T. e. indeterminate. 



It would be easy to show that the process adopted by M. 

 Poisson, when properly modified, as it must be to render it in 

 any way applicable, leads to the same result. 



It may be contended, that although we cannot follow out 

 the process when 5" = 0, yet the results must still hold; for 

 that as they hold for all finite and positive values of </, how- 

 ever small, they must also hold in the limit when q — 0. But 

 to this it may be answered, that we might as well consider 

 the definite integrals 



/ cosri/, / sinr^, 



to be the limits of the integrals 



/ 6~9y yP cos ry, I ^~9y yP s\nry^ 



where q is negative, as. where that quantity is positive. But 

 when q is negative, it is evident from the formulaj (B.), which 

 still hold, that 



/ e~^y yP COS ri/, and / e~9yyP s'm ry, 



are infinite for all finite values of <7, which shows that the 

 above argument is wholly untenable. The fact is, that the 



