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



views, yet something may perchance be said which will in abler hands be made useful in settling 

 some of the difficulties which beset the consideration of this perplexing subject. 



1. The ground on which I would reject the use of non-convergent series is a conviction 

 that such series may have some algebraical properties which their invelopments possess not, and 

 may lack others which the invelopments have. For series of ordinary forms I think 1 shall be able 

 to prove the truth of this as satisfactorily as such an intractable subject as an infinite non-converg- 

 jng series admits of. 



2. Let us notice first, that there is a presumptive ground of suspicion of the truth of this (viz., 

 that the algebraical properties of a non-convergent series are identical with those of its invelop- 

 ment) in the rejection of the anomalous term (the remainder) which if preserved would certainly 

 render their (numerical as well as) algebraical properties identical. Has the remainder wo alge- 

 braical properties ? If it has, then it will hardly be believed without proof, that in throwing 

 3t (and with it its properties) away we have not destroyed the algebraical equivalence which by its 

 means existed between the invelopment and the series. I will endeavour to illustrate my meaning 

 by instances. 



3. It admits of no doubt that including the remainder the equation ^=l—\+l — \ + 



ad inf. is strictly true. We are to examine whether this is algebraically true if the series be 

 taken without its remainder. Denote the sum of n terms of the series by S„ ; then it will be found 

 that for all values of n, S„= S\. This equation being strictly true may be made use of in any 

 algebraical operation : and as it is true however large be the value of n, it is impossible to refuse to 

 admit that S„ = S^ is a property of the infinite series. Hence 1, not being a root of this equation, 

 does not enjoy this property which the sum of the infinite series does enjoy, viz., that it is not 

 altered in value by being squared. ^ is the sum of the series inclusive of the remainder, and S^ is 

 the sum of the same series exclusive of the remainder. Hence the rejection of the remainder has 

 altered the algebraical properties of the symbol by which the series is represented. 



4. But the algebraical importance of the remainder may be rendered still more striking, and the 



a 

 b 



1-1-1+1 + .. .to a terms 



1 + 1 + 1 + ... to 6 terms 



1 + 1+1 + .. .to a terms 



impropriety of rejecting it put in a stronger view. For if any proper fraction — be put in the form 

 , it will be found by the ordinary process of algebraical division that 



1 + 1 + 1 + ... to6 terms 



= 1 - 1 + 1 - 1 + 



Now many persons have found it difficult to reject ^ as the algebraical equivalent of 1 - 1 + 1 - ... 

 because by ordinary algebraical development this series ad injinitum can be obtained from ^ . 

 It is here shewn however that the very same process which elicits the series from i would serve to 

 elicit it from any proper fraction whatever : and this being so, by what distinguishing property 

 are we to be guided, so as to be able to select amongst all proper fractions some one particular 

 value as the equivalent, the unique equivalent of the infinite series ? If 1 be selected as embody- 

 ing all the algebraical properties of the series, surely we must admit that for as good a reason 



- embodies the whole of its properties ; and thence we cannot avoid allowing that -^ and — are in an 



algebraical sense equivalent fractions. 



5. But it is said in special favor of \ that from whatsoever more general series 1 - 1 + 1 - ... 

 be deduced the symbolical equivalent is always found to be ^. If deduced, for example, from 



