364 On Newton's Rule for Imaginary Roots. 



secutive fulfilments of that condition, only one pair of imaginary 

 roots can with certainty be inferred. 



I had always regarded this first part of Newton's theorem as 

 the part which it was the more difficult to prove; and as 

 to the second part, namely, that a fulfilment after a failure 

 always implied a distinct imaginary pair, I conceived that a few 

 general considerations, such as those I have given, would suffice 

 to establish it. In this I was mistaken ; and I hasten to acknow- 

 ledge that my investigation is incomplete, and to at once with- 

 draw all claim to having furnished a complete demonstration of 

 Newton's Rule. 



Professor Sylvester's demonstration (in extenso) has not yet 

 appeared; but I have no doubt that it will be free from the 

 defect which attaches to mine. To Professor Sylvester alone 

 therefore the claim of priority will be due ; and I much regret 

 that by disputing this claim I should have put him to any trouble. 

 I cannot atone for it in any other way but this. He will see 

 that I have not allowed any considerations whatever to prevail 

 with me against my own conscientious convictions — against a 

 clear sense of what is just and right ; and I trust that this amende, 

 which is entirely spontaneous, and not the result of any external 

 influence, will be accepted in the same spirit in which it is ten- 

 dered. 



I remain, Gentlemen, 



Your obedient Servant, 



J. R. Young. 



P.S. — This letter would no doubt have been forwarded earlier, 

 but you are aware that my copy of the Magazine was sent by 

 post from your office (at my request) to another person : it did 

 not reach me till the 10th inst. 



[Mr. Sylvester, to whom, at Professor Young's request, the 

 above has been shown, has requested insertion of the following 

 observations. — Ed.] 



There can be but one opinion of the course adopted by Pro- 

 fessor Young in avowing his mistake ; it can only tend to raise 

 him in the estimation of all right-thinking persons. Most 

 writers in the course of their lives have published some erroneous 

 results, but comparatively few have had the magnanimity to 

 make so prompt an avowal. I wish it were not indispensable 

 for me to observe on the two charges of inaccurate statement 

 of matters of fact brought against me in the last Number of this 



