292 Prof. Young on Newton's Rule for Imaginary Roots. 



careful reader of my demonstration, all this must be abundantly 

 apparent. 



2. If, however, from having but an infirm grasp of the sub- 

 ject, anyone should demur to the above conclusion, or to that 

 marked 3 at page 116 of my former paper, let him further con- 

 sider that if only the three leading terms of an equation of the 

 nth degree be written down, that equation will necessarily have 

 two significant roots, the following coefficients being merely 

 zeros; that the introduction of another significant term will 

 cause the entrance of another significant root, of two significant 

 terms, two more significant roots, and so on ; that is, that for 

 k + l terms there must be k significant roots in the equation, 

 and that, too, how many soever of the terms between the first 

 and last (these themselves being significant terms) are replaced 

 by zeros. But if the pairs I,, I 2 , noted above, could be but one 

 and the same pair, then for £+1 terms there would be fewer 

 than k significant roots, which is impossible. 



I here take my leave of this subject — a subject on which 

 quite enough has, I think, now been said, and about which a 

 fuss, certainly more than enough, has been made. Professor 

 Sylvester's demonstration will very likely be preferable to mine 

 — preferable on the score of brevity, or of elementary simplicity, 

 or of lucid exposition. But whatever may be the merit of his 

 " discovery " in these respects, the merit (such as it is) of priority 

 belongs to me. 



Note. — In Professor Sylvester's communication in the last 

 Number of this Journal, there are two mistakes which ought to 

 be corrected. 



(1) It is said that I invited Professor Sylvester to express an 

 opinion on my supposed demonstration. I did nothing of the 

 kind. 



(2) Professor Sylvester says, "I caused to be forwarded to 

 Professor Young an invitation to attend rny lecture." The only 

 invitation that reached me was through f The Times ' newspaper, 

 on the afternoon of the day on which the lecture was to be 

 delivered. The former invitation was never given, the latter 

 was never received *, 



I crave permission to add to this paper two theorems, not 

 generally known, which may be found useful for determining 

 the character of the roots of equations of the third and fourth 

 degrees ; they were, I believe, first investigated in my { Course of 

 Mathematics/ and are as follows : — 



* I have been informed, however, that a copy of the Syllabus of the lec- 

 ture was left for me at the office of this Journal. 



