iW Mr»Ji^i Sylvester on et remarkable Discovery m the 



caring to calculate /,^*, it is enough for our present purpose to 

 satisfy ourselves that ^ cannot be zeix)^ as then the product 

 would have a factor (a-f/3 + 7)*. Hence^ then, on putting 

 a=bc, ff=^ac, y=ab, we see that the discriminant, when m is 0> 

 will be of the form 



But when »^ is (X H^ vanishes, and there is no term Si or ^3 in 

 Hg. Hence evidently the discriminant Hjq just found cannot 

 be dependent on Hg, H4, or Hgj nor is it possible to make 

 H,o+;jIV + (7H/.H6, i.e. (p + l)s^^+A^ .s^^-hffSs^ .s^ a per- 

 feet square on account of (/ not vanishing ; so there is no Hg 

 upon which Hjo can depend. Hence, admitting, as there seems 

 every reason to do, that the number of invariants of a function 

 oix, y of the degree m is m— 2, we find that the four invariants 

 in the case of the first degree are respectively of the second, 

 fourth, sixth, and tenth dimensions, a determination in itself as 

 a step to the completion of the theory of invariants of no minor 

 importance. 



But it seems hopeless by means of these forms to arrive at the 

 desired canonical reduction. The forms, however, of ^i ^g s^ are 

 very remarkable as not rising above the 1st, 1st and 2ud de- 

 grees respectively in s^^ s^, %. Also H4 vanishes when m=0 and. 

 H4 has been obtained by putting 



a .x^ + b7/'i-c:^-\- mouhjH^ 

 under the form of 



H5^ + 6B^y + 15Ca?y + SODci^^/^ ^ i5E^y 4. 6F^ + G?/«, 

 and taking the determinant 



Consequently in general the vanishing of the above-written de- 

 terminant will express the condition that a function of the sixth 

 degree may be decomposable into three sixth powers. This also 

 is true more generally. If F(^, y) be a function of 2i dimen- 

 sions, the vanishing of the resultant in respect to ^•*, a?*~^ .y, 

 ;': .y', (taken dialytically) of^^' '" '^^ 



\dx) ' ' \da;J 'dy'^' " ' \dy) '^ 

 will mdicate that F admits of being decomposed into i powers of 

 linear functions of x^ y\. 



*/=-625 ^=3125. 



t Such a function so decomposable may be termed meio-catalectic. Meio- 

 catalecticism for even-degreed fuuctions is the analogue of singularity for 

 odd-degreed functions. 



