of the Sums of Neutral Series. 143 



1 — x + x* — a, 3 + &c. m — . 



2 



It may be made to differ from that quantity by a quantity less 

 than any that can be assigned, but it is never actually to it — 

 least of* all is it so when x = 0, in which case it assumes a to- 

 tally different value. 



It was not for nothing that Newton devised his method of 

 limits, and to the present case it applies with peculiar clear- 

 ness and beauty. But no longer to fight with shadows, I 

 shall take up a definite position, and shall leave it to Pro- 

 fessor Young to drive me from it if he can. I assert, then, 

 that the series 



l-(l-#) + (l-#) 2 -(l-*) 3 +&c, 



so long as x lies between and 1, and differs from each of 

 them by an actual magnitude (I do not say a sensible magni- 

 tude, for the present is not a question of degree), approaches 



to — as its limit when x is made to diminish ; that when #=0, 

 2 



the absolute value of the series is 1 or indifferently ; that 



when x is less than 0, it becomes + co indifferently ; and I 



defy any man now living, or as the lawyers say, who may 



hereafter come in esse, to prove anything else, be it more or 



less concerning it. 



Prof. Young considers that the conclusion, that "the ex- 

 treme of the divergent cases is really infinite," could " never 

 have been anticipated from the theory hitherto prevalent." 

 Protesting as I do against the use of the term " extreme of 

 the divergent cases," I may say that long before Prof. Young 

 either said or wrote a word upon the subject, I had shown 

 that all the divergent cases have the value + co indifferently. 



Again, Prof. Young says, "if he has been anticipated in any 

 of these views, which are doubtless calculated to produce a 

 reform in the existing theory, he hopes to be informed of the 

 circumstance through the medium of this Journal." I beg to 

 assure him therefore, through the pages of this Journal, that 

 all his views which are not erroneous (though what propor- 

 tion that may be I confess myself unable to state, as I do not 

 understand very clearly what they are) have been anticipated 

 in my paper dated March 17, 1845, and published in the 

 number of this Journal for June in the past year. 



I now await Prof. Young's answer, trusting that I may not 

 be under the necessity of replying to any more of his papers 

 till he has had an opportunity of reading some of mine. 



January 3, 1846. 



