244 
ON THE SEXTAOTIC POINTS OF A PLANE CURVE. 
[341 
48. Now 
Jac. ( U, H, d x H) — g ^ g J ac - ( U, xd x ll + yd y H + zd z H, d x H), 
and the last-mentioned Jacobian is 
= 3 X H Jac. (U, x, 3 x H) + d y H Jac. (U, y, d x H) + d z H Jac. {TJ, z, d x H) 
+ y Jac. ( U, d y H, d x H) + z Jac. ( U, 3 Z H, 3 X H), 
where the second line is 
= — i/Jac. (U, d x H, d y H) + z Jac. {U, 3 Z H, d x H), 
or writing (A', B', G') for the first differential coefficients and (a', b', c', /', g', h') for 
the second differential coefficients of H, this is 
y 
A, B, G 
A, B, G 
a, h\ g' 
9 > f> o' 
h', b', f 
a', h', g' 
= B, C) + z(%, 33', 8X4, B, C). 
The first line is 
= A, B, C 
A', B', O' 
a!, h', g 
A {Eg - C'h') + B {GW - A'g') + C {AW - B'a), 
or reducing by the formulae, 
(3m - 7) (A', B', C) = {a'x + h'y + g'z, h'x + b'y +fz, g'x +f'y + c'z), 
= {A (- ®'y + $'*) + B (- %'y + S'*) + C (- G'y+S'*)) 
= 3^7 1_ÿ( ®'’ + S '’ B ’ °)1- 
Hence we have 
Jac. (U, H, d„H) = (l + gA^) {-y(@', S', 5. <7) + *($', S', s'$A b, 0)1 
1- y (©'. S', G'$-4, B’ C)+Z(*'. S', g'$yl, 5. (7)) ; 
(-*(«', &, ®'$A B, C)+x(®\ S', ®'$A B, 0)}, 
{-«($', S', S' B, C) + ÿ(5l', @'5^, B, C)). 
1 
3m — 7 
and in like manner 
Jac. ( U, H, d y H) = _ y 
Jac. (¡7, H, 3 2 H) = 7