Theory of Canonical Forms and of HyperdetermmanUi 40^ 



contain c^ and k^j but Hg can only contain as to these letters the 

 combination c^. 1^ ] hence r = 0. 



Consequently Hg does not depend on Hg, H4, Hg. As regards 

 Hgj H3, H4, H5, Hg not vanishing, this may be made at once 

 apparent by making all the letters but g vanish ; the H's then 

 become identical with the coefficients of 



none of which are zero except that of X^. The same or a similar 

 demonstration may be extended to H7 and easily generalized ; 

 hence, then, this most unexpected and surprising law is fully 

 made out*. 



To return to the subject of canonical forms, I have not found 

 the method so signally successful in its application to the 4th 

 and 8th degree, conduct to the solution of other degrees, such 

 as the 6th, 12th, or 16th, of all of which I have made trial ; pos- 

 sibly another canonical form must be substituted to meet the 

 exigency of these cases f ; and it may be remarked in general, that 

 if we have a function of the (2?i)th degree, the canonical form 

 assumed may be taken, 



where V, in lieu of being the squared product of 



{Pix + qiy){p^a?+q^) . . . {pn+i.^ + Qn+i-y) 



* This demonstration, however, does not extend to show that the coeffi- 

 cients of the powers of X may not possibly be dependents, i. e. exphcit 

 functions of one another combined with other invariants not included among 

 their number, or of these latter alone. For example, in the case of the 

 12th degree, we know by Mr. Cayley's law that there must be two inva- 

 riants of the 4th order. Our determinant gives only one of these. Call 

 the other one K4 ; by the above reasoning it is not disproved but that we 

 may have 



H6=i?.H/+gH2.H4+rH32-h5H2.K4. 



I believe, however, that the H's may be demonstrated without much 

 difficulty to be primitive or fundamental invariants. The law of Mr. Cayley 

 here adverted to admits of being stated in the following terms : — The num- 

 ber of independent invariants of the 4th order belonging to a function of 

 X, y of the nth. degr-ce is equal to the number of solutions in integers (not 

 less than zero) of the equation 2x-{-3y=zn—S. Vide his memorable papei* 

 (in which several numerical errors occur against which the reader should 

 be cautioned) On Linear Transformations, vol. i. Camb. and Dub. Math. 

 Journ., new series. There is no great difficulty in showing, by aid of the 

 doctrine of symmetrical functions, that there can never be more than one 

 quadratic or one cubic invariant, and in what cases there is one or the 

 other, or each, to any given function of two variables. The general law, 

 however, for the number of invariants of any order other than 2, 3, 4, 

 remains to be made out, and is a great desideratum in the theory.of iiifitr 

 transformations. _ ^r» ;, 



t See the Postscript for a verification of this conjecture. *' ''^ ' '^^- 



