660 ME. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OF A PLANE CUEYE. 
And as %, differ only in respect of a numerical factor, any other factor that can be 
predicated of % may he affirmed of %!, and vice versd ; and similarly for the other pairs 
§ 3. Formula? of Reduction. 
The degree of the expressions (28) or (29) is 18w— 36; it remains to show that existence 
of certain extraneous factors, which when divided out will reduce the degree to 12 n — 27, 
and at the same time render the three forms identical. But before entering upon this, 
it will be convenient to premise the following formulae, the first group of which are easily 
verified. 
y7j —zY =3(w— 2 )Hm 1 
zX—x Z = 3(n— 2)Hy 
xY-yX=3(n-2)Hw 
yb x Z — zB^Y — (3n—7)up—(n—l)wp -\-3{n—2)FLu x 
ydyTi —zdyY =(3n—7)uq ~{n — 1 )vp -\-3{n— 2)Hw' 
y~b z Z — zd e Y =(3n—7)ur -(w-l)wp+3(rc-2)IR/ 
zb x X-x'b x Z =(3n—7)v]) -(n-l)uq + 3(n-2)Hw' ^ 
zbyX.—x'byZ =(3n—7)vq — ( n — 1 )vq +3 (n — 2)11?;, 
zB 2 X— x'dJZ =(3n—7)vr —(n—l)wq-\-3(n—2)11u! 
x~b^Y —yb z X=(3n—7) , wp—{n—\)ur -\-3{n— 2)HV 
x~b y Y —ydfL =(3n-7)wq—(n—l)vr-\-3(n— 2 )Hu' 
#B 2 Y —yd z X=(3n—7)wr — (n—l)wr + 3(n—2)Hw,. > 
And writing 
-P 1 =^> I +Yr'+Z 2 ' | 
-Q-Xr' +Yq 1 + Zf (32) 
— R^X#' + Yf-^-Zr^ j 
then also 
Y^Z-Zd,Y=-( i ?P+wP 1 ) ZB.X-XB^-feP+flPJ XB 2 Y — YB ^X = — (rP + wP, ) 1 
YB y Z-Zb y Y=-( i? Q-fwQ 1 ) ZB.X-XB^-^Q+uQ,) XB^-YB^-^Q+wQ,) 1(38) 
YB 2 Z-ZB 2 Y=-(^R+wR,) Zh 2 X— XB z Z= — (g'R+'yRj) XB 2 Y-YB 2 X=-(rR+wR 1 )J 
Moreover, writing with Professor Cayley, 
(& b, c, jr, 0 , i)(B„ b„ h z ) 2 H=o 
3$, C, f, 0, fc)(B„ B„ B.ft, B yQjj= . . , B s Qy= . . 
BA=(B,a B,£, B 2 c, Bjf, d x 0, BJ>)(B„ B„ dJH, B,Qh= • • 
