On Differentiation as applied to Periodic Series. 213 



The anonymous correspondent Jesuiticus in the last Num- 

 ber, refers to those analytical researches triumphantly in fa- 

 vour of the undulatory theory of light. I do not write to dis- 

 turb the philosophical opinions of Jesuiticus, but to remind 

 the readers of the Magazine where they will find the discus- 

 sion of the points referred to. 



XLI. On Differentiation as applied to Periodic Series : with 

 a Jew Remarks in reply to Mr. Moon. By J. R. Young, 

 Professor of Mathematics in Belfast College*. 



F in the general expression at p. 430 of my paper on 

 Periodic Series, in the last volume of this Journal, A be 

 made equal to —1, we shall have the identity 



1 a n t , « a o . COS (w + 1 ) fl + COS M 



— = cos0 — cos 2 9 + cos 3 9— &c. -\ „ ,' — — tt ; 



2 - 2(l+cos0) * 



and if we multiply this by d 0, and integrate, we shall further 

 have 



6 | J 



— = sin sin 2 -f — sin 3 — &c. 



2 2 ' 3 



1 



±f 



cos (n -f 1)0 + cos n , . 



2(1 + cos0) 



Now it is demonstrable, from other and independent princi- 

 ples, that, when n is infinite, the right-hand member of this 



equation, omitting the integral, is the true development of 

 t 

 — , for all values of not exceeding it. Hence we may infer 



Z 



that, for n =s oo , this integral is necessarily zero. If we sup- 

 press it therefore, we shall commit no error in the expression 





 for — ; but a very considerable error will be introduced if we 



S5 



attempt to derive from that expression, thus limited to the 

 particular case of n = oo , a series of other equations, by the 

 aid of differentiation, as is commonly done. If the evanescent 

 integral be restored, we may then apply the process of differ- 

 entiation as far as we please : our resulting equations will all 

 be identical equations ; holding, whatever be the value of w, 

 and supplying the necessary corrections of those erroneous 

 developments which, in the case of n = oo , are so commonly 

 met with in analysis. 



I have elsewhere observed that differentiation fails to be 

 applicable to the series 



* Communicated by the Author. 



