Theory of Canonical Forms and of Hyperdetervnmanfs'. 393 



The solution of the problem given by me in the paper before 

 alluded to presents itself under an apparently different and rather 

 less simple form. Thus^ in the case in question, we shall find 

 according to that solution, 



{Pv« + W) {P^ + M) {P^ + W) It 



s= a numerical multiple of the determinant 



ax + hy; hx+cy, cx+dy 



bx + cy; cx + dy; dx + ey 



cx + dy; dx+ey; ex-\-fy. 



The two determinants, however, are in fact identical, as is 

 easily verified, for the coefiicients of c^ and j/^ are manifestly 

 alike; and the coefiicient of x^y in the second form will be 

 made up of the three determinants, 



a h d 



h d d 



h c e 

 c d f 



h b c 

 c c d 

 d d e 



of which the latter two vanish, and the first is identical with the 

 coefiicient of x'^y in the first solution. The same thing is obvi- 

 ously true in regard of the coefficients of xy"^ in the two forms, 

 and a like method may be applied to show that in all cases the 

 determinant above given is identical with the determinant of my 

 former paper, viz. 



a^x-\-a^\ ac^+a^; ... aj^+an+iy 



a^x + a^; a^x + a^; ... an+iX + an+2y 



....... i^ri:i\ \ 



a^x-^-an+iy, «n+i^ + ««+2y; • • • «2»i^+«2«+iy. 



Thus, then, we see that for odd-degreed functions, the reduc- 

 tion to their canonical form of the sum of (/i+ 1) powers depends 

 upon the solution of one single equation of the (?i 4- l)th degree, 

 and can never be effected in more than one way. 



This new form of the resolving determinant affords a beautiful 

 criterion for a function of x, y of the degree 2/i + 1 being com- 

 posed of w instead of, as in general, {n + 1) powers. In order 

 that this may be the case, it is obvious that two conditions must 

 be satisfied; but I pointed out in my supplemental paper on 

 canonical forms, that all the coefficients of the resolving deter- 

 minant must vanish, which appears to give far too many con- 

 ditions. Thus, suppose we have 



ffa?7 + 7hx^y + ^Ica^y^ + 35^<2?y +35ea^^4+21/a?V+ 7gxy^+hx\ 



The conditions of catalecticism, i. e. of its being expressible 

 under the form of the sum of three (instead of, as in general, four) ^^ 



