MU. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OF A PLANE CTJEVE. 655 
whence writing 
<E>=(a, b, c,f, g, h)(u, (3, y) 2 , 
we may derive 
(n-l)X 2 =-^ + 2 to(Zlu-hW+<B>y)-x 2 <P, 
(n-iy 
(n— l)v 2 = — ^ 2 C-l-2^(#a+Jf/3+Cy)— z 2 <&, 
(n— l)p = — & 2 #+&z(i§a+3$0 + JV)+^(#a+ Jf/3+Cy) — 
(w— 1 )j>A =— + Jf/3 + Cy)+^(S a +H/3+#y)— 
(w — 1 )Kfb = — h 2 $ + fy(&a + W + 7 ) + Ml «+Bf3+Jy)— 
But, as will be found on calculating the expressions, 
(n— l)DX=^(9[a + ^/3 + #y)— x$>, 1 
(n— l)D i M-=^0|a+B^+4fy)— y®, 1 
(n—l)Uv =&(#a+4f/3-|-Cy)--:s< I>, J 
so that 
(w-l) 2 X 2 =-^a+2(w-i>nx+^, 
(n—lfgJ 1 = — § 2 ^3 + 2(w— l)yn(A-\-y 2 ®, 
(n— 1)V = — £ 2 C + 2(w— l)z CH -|-z 2 <I>, 
(w— l) 2 p =— &tf+(n— l){yUv + z np)+yz<i>, 
(n—Yfvk =— h 2 0-\-(n— l)(z Wk-\-xnv)-\-zxQ, 
(n-l)\gj = —l 2 $l+(n—l)(x Dp+y □ x)+^. , 
Hence, if m be the degree of V, 
( 9 ) 
• (10) 
( 11 ) 
( 12 ) 
(^-i) 2 {A 2 B 2 y+ iy /b 2 y+v 2 B 2 v+2(pB^ 2 y+^3,Y+^^y)} 
= -S 2 (<3, 33, C, f, 0, l)a, 3„ B JV+2( % -l)(m-l)(n^,V+ D^V+ D^ 2 V), 
whence, substituting in (7), and bearing in mind that 
(n— 1)S1w+D + ^w)=H^, 1 
(n—l)^u+Mv+fw)=B.y, i (13) 
(n— l)#M+Jf y + Cw)=H 2 , j 
we have 
(»- 1 ) ! (i + 2 ar) ( □ *3,v+ □ j»a,v+ O -3.-V) - 8*(a, Ji.i.jr. 6, ® )(3„ 3,, a,) 5 v= o. 
But 
□ aV + □ i«aV + □aV=^(wDX+'yD 1 M/+^(;Dv) 
= , 7^1 ( (& u + D + + (Ifou + 33fl + fw)(3 + {<&u + tfv + €w)y } 
4x2 
