OF AN INFINITE ANGLE. 257 



and according to the definition of =, the other member is or ought to be the result of that opera- 

 tion. But we observe that 1 — x + x^ — is a scries of terms following the same law throush- 



out, and shewing no indication of any terms which are not included in this law ; yet it may 

 be asked, have we any just ground for knowing that all the terms resulting from the division 

 of 1 by 1 + .J7 rfo follow the same law throughout ? Let us examine ; if we stop after one term 



1 .r 

 of the quotient we find = 1 ; if we pursue the division a step further we find 



1 iC^ 1 v^ 



=1 - X + ; another step gives = I - x + ae^ '- , and so on. In all these 



\ + X \ + X \ + X i + w 



partial operations we observe that one term of the quotient is an exception to the law followed by 

 the others. It is true, by continuing the process, we may push this anomalous term to any con- 

 ceivable distance from the beginning of the series, but there is not the slightest indication that 

 by so pushing it it will at length cease to be, or become zero : on the contrary, as Professor De 

 Morgan justly remarks, by the prolongation of the operation it is removed farther off but not 



destroyed. Consequently, the operation represented by is of a character which can never 



be completely comprehended in any series of terms which follow one law : and thei'efore, strictly 

 speaking, there is no such quantity as the definition requires which can be joined with it by the 

 symbol =. Shall we then join it with as much of the quotient as does follow a fixed law? 

 It is clear we cannot without violating the terms of the definition. When therefore we find 



= \ - X + X- — ... ad inf. without an implied remainder, we are at a loss to understand in what 



1 -I- .r 



way this use of it is reconciled with the meaning attached to the sign = in the definition. Yet 



it is certain, that most eminent writers do use the symbol = to connect a function with a series 



every term of which is supposed to follow a fixed law, as though the operation denoted by the 



function were capable of being represented by such a series of terms. Still, though it is thus 



rendered evident that the usage has not been sanctioned by the definition, the discrepancy is 



not very important in itself, seeing that an alteration may be admitted into the definition which 



shall make it agree with usage. The definition may then stand thus ; — the sign = is used to 



I'onnevt an involved expression with the result of an operation as far as it is expressible in 



terms ivliich follow a fixed law. The really important point now to be examined is, whether 



that portion of a result herein included will in all cases represent, for algebraical purposes, the 



properties of the expression from which it was derived. If it will so represent the expression, 



then for algebraical purposes series of all kinds, whether convergent, periodic, or divergent, will 



stand on the same basis, and their use in all cases be equally safe. I need hardly say that 



this is a much disputed point, which has been warmly attacked and defended. I am induced to 



venture into the field oil the side of the assailants from having observed that its advocates have 



defended the use of non-convergent series on grounds some of which are capable of being easily 



shewn to be fallacious : and though I cannot bind myself to the justness of all the arguments 



which have been opposed to them even by the most eminent and skilful analysts, I yet think there 



are sufficient reasons left to justify us in rejecting non-convergent series when in accordance with 



the above definition their remainders are thrown away. 



Now according to the definition above proposed, it is evident that an invelopment and its series 

 are not efjunl, (they difi'er by the remainder) the question is, are they equivalent ? does the series 

 embody all the algebraical properties of the invelopment, and no more.'' The discussions which 

 have been so earnestly carried on with the view of arriving at a satisfactory settlement of this 

 difficulty have not yet elicited any unanswerable arguments on either side : at any rate they have 

 not been of such a character as to set the question at rest. Though I dp not presume to hope that 

 what is here brought forward will have the effect of satisfying those who entertain the opposite 



Vol. VIII. 1'aet III. L l 



