214- On Differentiation as applied to Periodic Series. 



a 111 



— = sin — — sin 2 9 + — sin 3 r sin 4 + . . . . 



2 2 3 4 



in the isolated case of n == oo ; and it is plain that in this case 



— sinra0 is 0, though sinw0 is itself indeterminate ; the inde- 



terminateness is therefore rendered nugatory. But if differ- 

 entiation be allowed, this indeterminateness reappears in 

 cos ii 0, in the right-hand member of the result, though the 

 left-hand member remains determinate, which is absurd. 

 Still we are not precluded from applying differentiation to the 

 general forms above, since these are universally true ; they 

 comprehend all values of w, and are identical. It is in virtue 

 of this identity, and of this alone, that the results of differen- 

 tiation may safely be extended to n = oo , although for this 

 isolated value of n differentiation be inapplicable. 



I have very little to say in reference to Mr. Moon's attack 

 in the last Number of this Magazine. The papers which have 

 called it forth, — whether justly or not, I leave others to deter- 

 mine, — Mr. Moon confesses that he does not understand ; 

 and humiliating as such a confession may seem, the whole 

 tenor of his remarks shows that he is sincere. 



I beg to say, that I did not write expressly for Mr. Moon, 

 and Mr. Moon therefore cannot reasonably expect that I 

 should attend to his demand, and define the terms I use. I 

 have employed nothing but the recognised language of ana- 

 lysis, and I cannot undertake to encumber the pages of this 

 Journal with a glossary of scientific terms for Mr. Moon's 

 especial benefit; if he will only take the trouble to turn to 

 the Penny Cyclopaedia, Mr. De Morgan will fully instruct 

 him in all these things. 



The occasion of my mentioning Mr. Moon's name was this : 

 — I found Mr. Moon, in the June Number of this Magazine, 

 floundering amidst difficulties which he showed himself un- 

 able to cope with. I had long previously contemplated a 

 paper, of which the main object was to remove those difficul- 

 ties, and in drawing it up for this Journal, I could not well 

 avoid the mention of Mr. Moon's name. But I mentioned 

 it with the most scrupulous courtesy and i*espect; I was espe- 

 cially anxious on this point, on account of the peculiarities 

 which Mr. Moon had so often displayed in his published 

 communications; so anxious indeed was I to avoid offence, 

 that — at the risk of losing all credit for discrimination — I even 

 went the length of calling him "an able contributor to this 

 Journal !" As I have already said, I did not certainly write 



