Prof. Young on the Theory of Conjugate Points. 9! 



It was my intention to have instituted a comparative series 

 of experiments witli the Hght obtained by ignition of the metals 

 by the voltaic battery, but this point has already, I find, en- 

 gaged the attention of my friend and colleague Professor 

 Wheatstone ; and he has made it the subject of an investiga- 

 tion, characterized by his usual ingenuity and accuracy. .^; 



XVI. On Imaginary Zeros, and the Theory of Conjugate 

 Poifits. By J. R. Young, Professor of Mathematics in 

 Belfast College^. 



IN a paper read at the last meeting of the British Associarr 

 tion, the writer of these remarks had occasion to notice' 

 the errors frequently committed by neglecting the algebraic 

 sign of zero in the extreme or limiting cases of certain infinite 

 series ; and to shov/ that this neglect had led Abel and other 

 distinguished analysts to conclusions incompatible with the 

 general laws of algebra. r 



For example: Abel says, in reference to the series ,^7 



1 . 1 . ^ , 1 . „ rsdjonA 



~^=sm(p- — sm2^+--sm39 -...., ^ ,„ „ ->-, 



that " lorsque <p = 7r ou — tt, la serie se reduit a zero, comme 

 on voit aisement. II suit de la, que la fonction '," ; / 



. J : 311} bei-(fu<. 



-It'll.} J sin (p — sin 2 (p + — sin 3 (p --^lir.^i^Kon liKO'i/iri') 



•>,:,,!:■; ! ,, , ^ ^ '--' :^'- v -kMJd 3run!' 



a la propriete remarquable, pour les valeurs ip =7r, et ^ = — tt 



d'etre discontinue." {(EuvreSf tom. i. p. 90.) 



But this conclusion is erroneous; for it is overlooked that 

 when <p becomes tt, simp vanishes positively, sin 2 ip negatively, 

 sin 3 (p positively, and so on ; and that when <p becomes — tt, 

 the signs of the several vanishing terms are the opposites of 

 these; so that, the continuity being still preserved, we have^ 

 in these extreme cases, uxlhrr- baJbmt .vmQjvT 

 1 /, 1 1 \ ^ • ' ---*- 



and 



yJOiiid boJiUilii 



from either of which we learn that 

 1 



T'' , 1 1 ., .„ 



=00=14-- 1 f- » ' ' '^'^' 



2 3 ' fi! lo noiov 



a result that we know to be true from other principles. 



* Read at the Meeting of the British Association at Cambridge, June 

 1845, and communicated by the author. 



