472 Mr. J. J. Sylvester on Inverse Orthogonalism 



of columns of all kinds, will be 



2({ (1 +*j/*i) (1 + VJ • • • (1 + -Vi) } • fr*w A*bi • • • i f*i)* 

 where the \ system is given, but the juu system is variable. 



But any factor (1 -\-X q . fi q ) is zero unless X q = /jb q . Hence for 

 any system of values of jjl not coincident with the A, system, the 

 corresponding multiplier of (ji v fi 2i . , . , fa) vanishes, and for 

 that system it becomes 2*. Hence 



E = 2'(\„ A 2> . . . ,Xj), 

 as was to be proved. 



12. These formulas admit of a useful application to Newton's 

 rule. 



The two superior limits to the number of roots included be- 

 tween (a) and (b) which it (or my extension of it) furnished are 

 A(+ +) and — A(H — ), where A refers to the ascent from a 

 to b, and the series are those mentioned in art. 9. Hence, 

 calling the two limits \, \', remembering that s is constant, 



As, +As 2 +As 1)2 



A= 3 



4 



, As, — As 2 + As 1>2 . 



K = t 



4 

 so that the limits are 



i{A Sl +A s , ;2 }±^. 



The mean of these is |(As, — As lt 2 ), which a fortiori is also a 

 superior limit. Here s l refers to the series 



J>Jv J<2> • • • JJn> 



and s h 2 refers to the series 



/G, /,G„ / 2 G 2 , . . . ,/„G n , 



which I have called, in an article in this Magazine, the H series. 

 If p is the number of permanencies in the f and </> in the H 

 series, it is readily seen that 



As, + As,, 2 _ A/+A<ft 

 4 " 2 



Hence, since X and V are each of them superior limits, it fol- 

 lows as an immediate consequence that — - is so likewise ; 



but this assertion conveys no new information, and ought not to be 

 treated as a new theorem, as I inadvertently stated it to be ; the 

 fact, however, of its being implied in what was previously known 

 is so far from being immediately evident, that M. Angelo 



