40 



small a value we attribute to t, the series must always diverge 

 after a certain number of terms."* 



The reasoning by which a finite sum is determined for s, 

 when ^ = 1, is as follows : 



sdt 



— = I. dt-l. 2 tdt+\. 2.3 fdt-8ic. (7) 



.-. ^^=^-l.«Hl.2<^- &c. (8) 



(9) 





:t— St 







sdt 



T 



= (1- 



■ s)dt - 



- tds, 



• 



ds 1 

 dt'^~ 





1 

 -J' 



and from this is found, for s, the definite integral 



-e \ e 'dt; 



t %J0 



from which it is inferred that " if ^ = 1, or the above integral 

 be taken from ^ = to ^ = 1, we have the expression for the 

 value of the series 



1 - 1.2 + 1.2.3 — &c." 

 Now several objections lie against the preceding reasoning : 

 in the first place it is assumed, in the final step, that « vanishes, 

 for t-=zO, notwithstanding that " however small a value we 

 attribute to t the series must always diverge," and thus at 

 length furnish terms infinitely great : and in the next place it 

 is assumed — and the assumption is somewhat similar to that 



* If, however, t be indefinitely near to zero, the " certain number of 

 terms" adverted to in the text, will be indefinitely great ; that is, the diver- 

 gency will be indefinitely postponed : the series therefore cannot be consi- 

 dered as divergent up to the limit < = 0; yet, as the statement in the text 

 seems to imply this, I have considered it to be comprehended in the hypo- 

 thesis ; although, as I have shewn, the point is of uo moment in the matter 

 under discussion. 



