894 Mr. J. J. Sylvester on a remarkable Discovenj in t^^ 



seventh powers, requires that all the coefficients of the different 

 powers of ar and p must vanish in the determinant 



a 

 b 

 c 

 d 



c 

 d 



f 



d e 



e f 



f 9 



9 h\ 



in other words, we must have five determinants, 



d 



6 

 f 



9 

 d 

 e 



f 

 9 



c 

 d 

 d 

 e 



c 

 d 

 e 



f 



f 



9 



d 

 e 



f 

 9 



e 



f 

 9 

 h 



e 



f 



9 

 h. 



all separately zero. But by my homoloidal law, all these five 

 equations amount only (5— 4) (5— 3), i. e. to 2. I may notice 

 here, that a theorem substantially identical with this law, and 

 another absolutely identical with the theorem of compound de- 

 terminants given by me in this Magazine, and afterwards gene- 

 ralized in a paper also published in this Magazine, entitled '' On 

 the Relations between the Minor Determinants of Linearly 

 Equivalent Quadi-atic Forms," have been subsequently published 

 as original in a recent number of M. Liouville's journal. 



The general condition of mere singularity, as distinguished 

 from catalecticism, i. e. of the function of the degree 2n + 1, being 

 incapable of being expressed as the sum of 2n + 1 powers, is that 

 the resolving resultant shall have two equal roots ; in other 

 words, that its determinant shall be zero, which will be expressed 

 b^ an equation of 2n(n+l) dimensions in respect of the coeffi- 

 cients. Mr. Cayley has pointed out to me a very elegant mode 

 of identifying the two forms of the resolving resultant, which I 

 have much pleasure in subjoining. 1?ake as the example a func- 

 tion of the fifth degree, we have by the multiplication of deter- 

 minants, 



