Theory of Canonical Forms and of Hyperdeterminanis, 403 

 Accordingly, by eliminating >/ v> > 



'^ mif^3-i 3kiA ,^iH? ,YRS o) «t ^m\i. 

 we obtain as the required resultant,^- '- -^ \t . « ir.'! ajfi/^ snla/ 



A tc sij^/ofi i)dt io ai^sioSii'JO'j'ofB'im^eoii^ '1_^ . 7'; ^ fuuexf / 



.liii 



.11 r 



taJ .(^^ ^;f,) Io ^itnoiorflooi odi'bxm v/Clb mn'i'2 in JiiiiJlijaai Biiii 



-v(i)-' 



Inasmucb as all the coefficients of \ in this expression are in- 

 variants of f{x, 7J), and these are the invariants of the first 

 order, it is clear that the coefficient of X* must be always zero, 

 which is easily verified. 



Again, if l is odd, the determinant remains unaltered if we 

 write —X for \; hence when/(^, y) is of the degree 46 + 2, all 

 the coefficients of the odd powers of \ disappear. Thus, then, 

 our theorem at once demonstrates that a function of x, y of the 

 degree 46 has 26 invariants of all degrees from 2 up to 26 + 1 

 inclusive, and that a function of ocj y of the degree 46 + 2 has 

 6 + 1 invariants whose degrees correspond to all the even num- 

 bers in the series from 2 to 26 + 2. 



But in order that the proposition, as above stated, may be 

 understood in its full import and value, it is necessary to show 

 that these invariants are independent of one another, which is 

 usually a most troublesome and difficult task in inquiries of this 

 description, but which the peculiar form of our grand deter- 

 minant enables us to accomplish with extraordinary facility. In 

 order to make the spirit of the demonstration more apparent, 

 take the case of a function of the twelfth degree, whose coeffi- 

 cients, divided by the successive binomial numbers 1, 12, 



— ^ — -, &c, may be called ^ 



fl, 6, c, d, e,f, g, h, l,j, k, /, m. S>. 



* Mr. Cayley has made the valuable observation, that X (given by equa- 

 ting to zero the above determinant) may be defined by means of the equation 



(^ being itself a certain rational integral form of a function of the ith degree, 

 the ratio of whose coefficients would be given by virtue of the above equa- 

 tions as functions of X and the coefficients off{x,j/). , v ^ 



S ."jXJl-i' .aid T '&,^ l" ~' — ~o — r — -*-|-^»''^ 



