ME. W. SPOTTISWOODE ON THE CONTACT OE CIJEVES. 49 
Proceeding to the calculation of the final expressions for the coefficients 5, c, /, g, h 
of the conic of five-pointed contact, which is given rather more in detail, we have, 
d\5 dV d\] 
y^, 2zx, 2xy=.T>^x\y^, 2 ^, 2zx, 2xy) 
+ 0 ^( 2 ^ 2zx, 2xy, x^) 
+I>^z%2zx, 2xy, x^ if) 
-\-2T>^zx{2xy^ x^^ y^^ z^) 
y^, 2zx) 
z= — (S(J)ivy-\-4:wT>hv—y'Dhv)v (v D H— 3HD v ) 
-{-(3(Du )^-t-4y D^y + 2 D®y )iv{iv'D H— 3HD iv) 
— ( vTD^y )iy(wD 2 H — 3HD2W) 
— ( xDhv)v (y DiH— 3HDiW) 
= ^^v\Bvf-v‘^iBlvf)J)B.-\-4.vw(wm-vI)hv)J)I^ 
-f (zy-^D^y +y-^D®w)DH— ^(w^Dh’DaHd* 
-f 3H { 3D yD zy ( yD 2 y — wD y ) + 4y?y (D yD% — D^yD^y) 
—yvD yD^ zy — zzy D zyD^y -f .y yD jyD^zy + xiuTi^ivT^v } 
= 3('zyDy— yDzy)D(yzy)DH-f 4y^yD(^yDy— yDzy)DH 
— 9(iyDy— yDzy)DyDzyH+12yzy(DyD^w— DwD^y)H 
+ D^y { w^( 0 DH — yD 2 H) — 3Hzy(sDzy — yD 2 ?y} } 
+ Dhy { y^(j/DH - yD.H) - 3Hy(yD y - yD.y) } 
= (w-l)-V{3D(yzy)HDH+4yzy(DH)^-9DyDzyH^} 
+12 yzyH(D yD^zy — DzyD^y) 
+ 3yzyHD®(yDzy— zyDy) 
— 3yzyH(DyD^zy— DzyD^y), 
which, omitting the common factor (zz— 1)"V, 
= 3D(yw)HDH+4yzy(DH)^+9(?z-l)-VH^,^+9yzy9[H^ 
MDCCCLXII. 
H 
