IN REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 439 



Of the two Notes which follow, the second has already been referred to, by anticipation, in the 

 text : the first is intended further to confirm and establish the accuracy of the general theory which 

 pervades this Paper. 



Note (A.) 



It appears from the preceding observations that in certain infinite series involving a quantity subject to con- 

 tinuous variation we are presented in the extreme or limiting cases, with instances of what may be called 

 insensible convergence and insensible divergenci/ . The peculiarity of such cases consists in this: — that, within 

 a finite extent ol' a certain infinite range of terms, the convergency or divergency of the series is insensible ; so 

 that for such a finite extent the series does not sensibly differ from what I have proposed to call a neutral or 

 independent series. When however we pass beyond this finite range, and in imagination contemplate the terms 

 infinitely remote, we at once recognize the accumulated effect of these insensible variations ; and the con- 

 vergency or divergency of the series becomes abundantly apparent. The infinitely remote term at which this 

 fact discovers itself, is alike the termination of one infinite range of terms and the commencement of another: 

 the completion of which, if the expression may be allowed, shows the effect of the insensible variations through 

 a second infinite range, and so on. 



We are thus unavoidably led to the contemplation of different orders of infinites and different order^ 

 of zeros — things altogether beyond the reach of actual ocular examination. But those who take that com- 

 prehensive view of the scope and powers of analysis, which its own well-established results, and the practice of 

 those most deeply imbued with its spirit so fully justify, will not, I think, found any objection to the reason- 

 ings in the foregoing Paper on this circumstance. In fact, in the common doctrine of vanishing fractions, the 



very same principles are virtually recognized : the symbols - and — , which ought perhaps rather to be 

 written — , and — ; , may each represent any ratio whatever: — even infinity: so that the reasonings adverted 



to involve in them nothing repugnant to generally received conclusions. The symbols here noticed, when 

 really determinate, are so solely in consequence of their being governed by the principle of continuity. 

 This is pretty generally admitted : but there are certain other results of analysis, which the same principle 

 equally controls, but over which its influence is little suspected. Every one admits the truth of the equation 

 «° = 1, whatever be the value of a, without any reference to the law of continuity : yet if we reason from this 

 equation — still keeping the conditions of continuity out of sight — we shall speedily be led to conclusions of a 

 very startling character, as follows : — 



1 

 a" = 1; .-. a = l° = 1*, 



that is to say, unity raised to the power infinity is equal to any quantity whatever ! 



Note (B.) 



It was observed at page 432 that every neutral converging series might, without error, be replaced by the 

 corresponding dependent series. This observation might have been rendered more comprehensive ; for 

 diverging series, whose terms continually tend to zero, might also have been included, since the dependent 

 series, corresponding to these, have infinite sums, as well as the independent diverging series themselves. 

 These infinites however are not .strictly identical in the two cases: and by saying that the one series may 

 replace the other, nothing more is meant than that the sum in either case will be infinite. I'oisson, Abel, and 

 others, have shown that 



- log ( 1 + 2 a cos ip + a') = a cos rp — - a' cos 2<)> + -a' cos 3<p - &c. 



Vol.. VIII. Pakt IV. 3L 



