- oCldH + 7/012 = W, 
[341 
(J) 
(J) 
form the 
V, H)H- 
341] ON THE SEXTACTJLC POINTS OF A PLANE CURVE. 229 
15. Hence the condition will contain the factor S-, and throwing out this, and 
also the constant factor ——- , it becomes 
m — 2 
(— 15 m 2 + 60m — 57) H Jac. (U, V , H) H 
+ (30m — 54) (m — 2) H Jac. (U, V H, H) 
+ (m — 2) 2 (9H' 2 dCl — 45 HildH + 4O' v F0jH) = 0. 
16. We have 
d x (VH) = (d x .^)H + d x vH > 
viz. in (d x . V ) //, treating V as a function of (x, y, z) we operate upon it with d x 
to obtain the new symbol 0^. V, and with this we operate on H; in 0^ V we simply 
multiply together the symbols d x and V, giving a new symbol of the form (cl x , d x d y , d x d z ) 
which then operates on H. We have the like values of 0 ?y (VU) and 0j.(ViJ); and 
thence also 
Jac. (U, VH, H) = Jac. (U, V, H) H + Jac. (U, VH, H), 
viz. in the determinant Jac. (U, V, H) the second line corresponding to V is 0*. V, 
dy.V, 0«-V (V being the operand); and the Jacobian thus obtained is a symbol 
which operates on H giving Jac. (U, V, H) II; and in the determinant Jac. (U, VH, H) 
the second line is d x V H, 0 ?/ VH, d z ^ H (V being simply multiplied by d x , d y , d z respec 
tively). 
17. Substituting, the condition becomes 
(— 15m 2 + 60m — 57) H Jac. (U, V, H) H 
+ (30m — 54) (m — 2) [H Jac. (27, V, H) H + Jac. (U, VJ, H)} 
+ (m - 2) 2 {9H 2 dn - 54HVLdH + 40^077} = 0, 
or, what is the same thing, 
(15m 2 — 54m 4- 51)H Jac. (TJ, V , H) H 
+ (30m — 54) (m — 2) H Jac. ( TJ, V H, H) 
+ (m — 2) 2 {9H' 2 d£l — 45HTddH + 404/077} = 0. 
Article Nos. 18 to 27.—Fourth transformation, and final form of the condition for a 
Sextactic Point. 
18. I write 
(5m — 12) H077 — (3m — 6) 77012 = ^ Jac. (U, 12, H) (J) 
12077 + 77012 = 0 (iiH), 
and, introducing for convenience the new symbol W,