242 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
We thus obtain 
(da,. .][x, y, zf (21', . /$>,..) 
+ (a, . .Job, y, zf (21', . .Jda, ..) 
— 2 (21',. .Jax + hy+gz,. Jxda + ydb + zdc,..) 
(a, ..Jx, y, zf(dW, . .Ja,..) 
— (921', . .Jax + hy 4- gz, . .) 2 
2 (a, . .Job, y, zjdx, dy, dz) (21', . .Ja, ..) 
— 2 (21', .Jax + hy+gz, Jadx + hdy + gdz, .) 
= (<A, • -Joe, y, zf (321,. .Ja,..) 
— (021, . .Ja'x + h'y + g'z, . .f, 
= (da',. .Jx, y, zf (21,. .Ja',..) 
+ (a',. .Jx, y, s) 2 (21,. .Jda'..) 
— 2(21,..]£«.'#+ h'y+g'z, ..Jxda' + ydh' + 20#',..), 
= 2(a, ..Jx, y, zjdx, dy, dz)(21, ..Ja', ..) 
— 2 (21, .Ja'x + h'y + g'z,. .Ja'dx+h'dy+g'dz,..). 
45. If in these equations respectively we suppose as before that (a, b, c, f, g, h) 
are the second differential coefficients of a function U of the order to, and (a', b', c', f, g', h') 
the second differential coefficients of a function U' of the order to' ; and that (A, B, G), 
(A', B', O') are the first differential coefficients of these functions respectively, then 
after some easy reductions we have 
(to - 1) (to - 2) 0U(21',. .Ja,..) = to'(to' -1) U' (021,..]{a,..) 
+ to (to — 1) U (21', . .Jda,..) 
-2(m-l)(m-2)(W,..JA,B,GJdA,dB,dC) 
m(m—l)U(dA\. .Ja,..) 
-(m-lf(dA',..JA, B, Gf 
2 (to— 1)00(21', ..Ja,..) 
- 2 (to - 1) (21', . .JA, B, CJdA, dB, dC) 
~(m'-lf(d%..JA', B', C'f, 
(to'-1)(to'-2)dU'(21, ..Ja',..) 
+ to'(to' — 1) O'(2l,..#0a', ..) 
- 2(to'-1)(to'-2)(21,5', C'JdA',dB',dC') 
2 (to' — 1)00' (21, . Afa',..) 
- 2(to'- 1)(21,..JA', B', G'JdA', dB', 0(5'), 
equations which may be verified by remarking that their sum is 
to (to- 1) {0O(21', ..$«, ..)+U[(21', . .£0a, ..) + (021', . .Ja,..)]} 
- (to - l) 2 |02l', ..JA, B, Gf + (21', .JA, B, CJdA, dB, 00)} = to'(to'- 1) &c., 
viz., this is the derivative with d of the equation 
to (to — 1) U (21',. Ja,..) — (to — l) 2 (A', . .$yl, B, Gf — to' (to' — 1) &c. 
46. Taking now U'— H, and therefore to' = 3to —6; putting also U=0, 00=0, 
and writing as before 
№ = (021,..$4', B', C'f, 
№ =( 21 ,.JA', B', G'JdA', dB, 00'), 
E^ = (dW,.JA, B, Gf, 
F^=( 21', ..$4, 5, O#0A 05, 00), 
Oil = (021,. Ja',..), 
.Oil =( 2l,..$0a',..),