260 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE 



that the terms of the series 1 ~ x + w" - ... at an infinite distance from the beginning do not 

 convero-e towards 1 as their limit, but to one of the indeterminate quantities e'" or e*" ; the values 

 of these depending upon the law under which x approaches unity. Who shall prescribe this law ? 

 Surely it is (and must be left) arbitrary in the fullest sense of the woi-d. It is not true then that 

 the converging and diverging forms of 1 - x + x' - ... approach the same form, viz. 1 - 1 + 1 - 1 + ... 

 ad infinitum, as their common limit. For the limiting forms both of convergency and diver- 

 gency are arbitrary, yet so restricted that they never can mutually approach so near as to be 

 separable by only a single form : for e"*"^ never can approach so near to e"" that only unity lies 

 between them, because y is necessarily ^«i<e, i.e. neither indefinitely large nor indefinitely small. 



9. The unavoidable inference from the last article is that 1 - 1 + 1 - ... is an isolated form 



of 1 - X + x"- — and separated from the limits of continuity on either side by a finite interval. 



For the same reason it is an isolated form of 1 - w" + x*" — x"*'' + x^'' - ... Let it now be admitted 



that is the equivalent of 1 - x + x^ - ... ad infinitum, then it will follow that 



1 + X 



limit of = limit of (l - a + x- - ...) 



1 +x 



But limit of (l - x + x" - ...) is not = 1 - 1 + 1 - 



.-. limit of is not = 1 — 1 + 1 — ... 



1 + X 



a 

 This then is the proof that ^ is not, (and in a similar way it would follow that — is not) 



the proper equivalent of 1 — 1+1- even assuming to be the proper equivalent of 



I •\- X 



1 ± x" 

 1 - X + x'^ — It is easily shewn, since = I — x + x"- - ... n terms, that ^ (1 =*= e*'') is 



= limit (1 - .r + x" - x' + ... ad infinitum), which is therefore indefinite. 



10. In a paper " On Divergent Series" by Mr. De Morgan, there is a remark which shews the 

 important bearing of the results obtained in the preceding articles. " It is clear enough," writes the 

 Professor, " from the manner in which Fourier, Poisson, Cauchy, &c. use the limiting form 

 1 — 1 +) — ... that they intend it to signify ^ in an absolute manner. The whole fabric of periodic 

 series and integrals, which all have had so much share in erecting, would fall instantly if it were 

 shewn to be possible that 1-1 + 1 - ... might be one quantity as a limiting form oi Ao-Ai + A^- ••■ 

 and another as a limiting form of B^- Bi + B.,- ...". I object, of course, to the assumption that 

 1 — 1 + 1 — ... is a limiting form of the series alluded to ; but passing over that, it is shewn above 

 that 1 - 1 + 1 - ... when taken as a form of 1 - x" + x'' - ..., which it certainly is, may be one 

 thing or another, according to the values arbitrarily assigned to a and b. Indeed it is stated 



in Woodhouse's Anal. Calc. p. 61, that 1-1 + 1-1+.. . = , as well as = . But 



^ 1+1+1 1+1 



Woodhouse either did not observe this evident contradiction, or must have got over it by the 

 mystical maxim that is not = ^, and is not = -^ ; which is perhaps the case, for in a 



note he considers that Euler, Leibnitz, and Waring had fallen into a mistake by making , 



&c. = ^ , -g^ , &c. However, passing by this doctrine, it serves the purpose for which I 



quote it, for it exhibits Woodhouse as testifying to the propriety of taking 1 - 1 + 1 - ... to be 

 a form of the series \ - x + x^ - x* + ... which arises from the expansion of j . In fact, 



