654 ME. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
eliminated) the equations W= 0, □ W = 0, □ 2 W = 0 are replaced by 
^W == VV = B f W = AW (4) 
where ts is a numerical factor, and 
a=(b, & c, #, e, ** ( 5 ) 
To this preliminary transformation the first section of the paper is devoted *. The 
second section contains the actual elimination of the constants of the conic, and the 
reduction of the resultant to six forms, 3£=0, JH=:0, =j^,=0, %!= 0, JH'=0, 
of which % and 01 and 4jW, and ffl! differ respectively only by one and the same 
numerical factor, viz. (n— 2) 3 . All these forms, however, contain extraneous factors, 
the determination of which is the object of the remainder of the paper. The third 
section is devoted to the establishment of some formulae of reduction, the demonstra- 
tions of which are rather too long to be conveniently inserted in what would otherwise 
be their more natural place (§ 4). Besides these I have established many others of a 
like nature ; but the specimens here given will doubtless suffice to suggest the mode of 
proof of the rest to any one desirous of pursuing the subject further. In the fourth 
section it is shown that all six forms are divisible by the Hessian of U, and 
that %, %! are also divisible by u 3 , 01, 0\! by v 3 , and by w 3 , and that the result 
of these divisions is a single expression of the degree Yin— 27. 
§ 1. Preliminary Transformation. 
The first two equations of the system (3) are, as is well known, equivalent to the 
following, viz. 
U V w 
( 6 ) 
where 0 is indeterminate. The third equation, viz. □ 2 V=0, when written in full, is 
o= □?a,v+ n^y+ □^ z y+x 2 h 2 y+^ 2 y+^ 2 Y+ 2 (p^,y+ABAV+^^v). (7) 
Noww being the degree of U, we may without difficulty establish the following formulae 
given by Cayley (Z. c . p. 381) : 
(n—l)u 2 =—$$z 2 -\-lJfzy—&y 2 , 
(n—l)v 2 = — €x 2 -j-2(Bxz— Qz 2 , 
(n—l)w 2 =— % 2 +■ 2^yx — B# 2 , 
(n — 1 )vw = — $x 2 — <Bxy —^xz-\- Qyz, 
{n—l )wu = — $yx + <% 2 — ^foyz -f- Y>zx, 
(n — 1 )vw = — fzx—<Bzy + $}z 2 -f- €xy. 
( 8 ) 
* In a paper recently published in the ‘ Quarterly Journal of Mathematics,’ vol. vii. p. 114, I have given a 
transformation having the same object in view ; but its form is partial and in some sense incomplete, and the 
mode of proof less direct and obvious than that given in the text. 
