ME. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OE A PLANE CTJEVE. 659 
which may be regarded as the three forms by any one of which □ 3 Y=0 may be 
replaced. Before proceeding farther, it will be convenient to notice that the quanti- 
ties uX— #P, . . are capable of being transformed in a manner which will be useful 
hereafter, as follows : — 
TlX = Xu x x + (ww' — vv')jjx -\-(v'q— w'r)ux 
=Xu 1 x + ( ww ' — vv’)( 3n—2ll—qg—rz)—(v'q— w'r)(vy — wz) 
= Xfax + w'y + v'z) + 3 (n — 2)H (ww' — vv') 
=(n—l)uX-\-3(n—2)~H.(ww'—vv'), 
i. e. 
— uX+x¥=(n— 2){uX— 3H(W — ww')} 1 
-wY+^P=(w-2){mY-3H(ww 1 -W )} l (26) 
— vJL -\-z¥=(n— 2){uZ — ■3H(W — vu x )}. J 
Returning to (25), and taking any one of the three as W, we shall have for □ 3 V=0, 
□ 4 V=0, □ 5 V=0, 
a~bJ(uX Kb x (uY —yF)-\-gd x (uZ — ^P) — 0 2 u =0 1 
dd y (uX-xF)-\-lib y (uY—yF)-{-gb y (uZ—zV) — 0 2 v =0 | 
ad z (uX—xV)-\-Jid z (uY—yV)-\-g'b s: (uZ—zF)— 0 2 w =0 
a A (uX — x~P) + A A(wY — yP) + gA(uZ-zT) - ^11= 0 ; , 
and similar groups may be formed from the other two equations of (25). Now as (27) 
contain only three out of the six constants a, . . f x . . , and the single indeterminate A,, they 
are sufficient for the elimination in view, and give for the equation whereby the sextactic 
points are to be determined, 
B,(mX-^P) 
B/«*Y-yP) 
B>Z-zP) a, 
d>X-tfP) 
B/*Y-yP) 
B/wZ— zP) u 
=0, j 
i 
B>X-#P) 
B/uY-yP) 
c) z (uZ—zP) w 
r 
J> 
1 
% 
A( W Y-yP) 
A(uZ-zF) „ 2 H 
i 
which, in virtue of (26), may also be written in the form 
(^{wX— 3H)w/— vow')} 'b x {uY—‘YSi(wu l —uv')} B*{>Z— 3H(W— vu x )} 
d y {wX— 3H)W— ww')} 'b y {uY—oH.(wu l —uv')} ~b y {uZ— 3H(W— vuj} 
B J .{wX— 3PI)W— ww’)} B z {wY— 3H(«nq — uv ')-} ~b z {uZ— 3H(W— vu x )} 
A{wX— 3H)W— ww 1 )} A{wY— 3H(mq— uv')} A{uZ— 3H(W— vu x )} 
= 0 , 
v 
w 
G3r 2 H 
with similar expressions in v, Q ; w, R. Calling (28) and (29) %, %' respectively,' we 
may designate the entire group of six forms, three of the form (28), and three of the 
form (29) by 
1=0, ifl=0, #=0, 31' =0, iH'=0, $,=0. 
(30) 
