4*90 Mr. Moon on the Symbols sin oo and cos oo, 



.-. — + A cos 9 + A2 cos 2 9 + &c. + A^^-^ cos (« — 1) 9 



__1 1 — A^ . coswfl — Acos (w — l)d 



"" 2 1 - 2 A cos 9 + hP- ""' l-2Acose+A2 * 



Now it might be supposed at first sight, that when n is in- 

 finite we should have cos n — \ .^ ■= cos n 6, and that conse- 

 quently the limit of cos w 9 — A cos w — 1 9 when A = 1 is 0. 

 But a very little consideration of the true nature of the func- 

 tion cos w 9 — cos n — I $, will show that such cannot be the 

 case. For that function simply expresses the difference of the 

 cosines of two infinite or (which in considering the value of 

 their cosines is the same thing) indeterminate arcs, all that 

 can be said respecting which is, that the difference of their 

 absolute magnitudes is the same constant quantity 9, and hence 

 that function may have any of the series of values assumed by 

 cos (a 4- fl) — cos a, where a increases gradually from up to 



6 d 



CO , i. e. may have any value between — 2 sin — and + 2 sin-—, 



whence it is evidently not true that when A = 1, the limit of 

 the series 



1 TT 2 TT 



-^ -I- A cos -^ (jr — o) + A^ cos -j- [x — v) + &c. in inf. 

 1 1-A2 



IS 



1— 2 Acos-^(j;— u) -f- A2 



and therefore Fourier's theorem falls to the ground. 



I might here go on to show that, assuming the general truth 

 of Fourier's theorem, it might be made use of to prove the 

 fallacy of Mr. De Morgan's interpretation of the above func- 

 tion, — / do) / cos u){v — w).vdv. For in the common 



method of applying Fourier's theorem to discontinuous func- 

 tions, it is a mistake to suppose that at the limits of discon- 

 tinuity the true value of the function is the mean between the 

 two adjacent values. For my present purpose, however, 

 enough has been said ; and as I believe Fourier's theorem to 

 be unfounded in fact, it would be to little purpose to correct 

 a mistake in its application *. 



Having thus, as I hope, exposed the fallacy of the argu- 

 ments adduced by Mr. De Morgan in order to impugn the 

 validity of the direct method of ascertaining the values of sin oo 



* As the error in question turns upon a curious point in analysis, I may 

 on a future occasion revert to it. 



