250 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
and substituting the values 
TU=2<P, Vtf=—— tf, BU=SH, 
to — 1 
we find the before-mentioned expression of di 2 U. 
60. Operating with the two sides of the same equation on a function H of the 
order to', we find 
to (in — 1) UrH — (in — l) 2 d' 2 H = m (in' — 1) A>H 
-2 (m'-l)$V£T 
+ &0&; 
and in particular if H is the Hessian, then writing m! = 3to — 6, and putting U = 0, 
we find the before-mentioned expression for d 2 H. 
61. But we may also from the general identical equation deduce the expression 
for (3Hf. In fact taking H a function of the degree in' and writing 
( a , fi, y) = (d x H, d y H, 3 Z H), 
we have 
m (in — 1) U (a, .. h)/xd z H — vd y H, vd x H — Xd z H, Xd y H — fid x H) 2 — (in — l) 2 (3H) 2 
= m'^H 2 - 2m'^HV H + ^ 2 (21, .. $3 X H, 3 y H, d z Hf; 
and if H be the Hessian, then writing in' = 3in — 6 and putting also 17= 0, we find 
the before-mentioned expression for (3H) 2 . 
62. Proof of equation 
^ = “ i (*&* + tfiv + Z dz) + —“i V • 
We have 
3 2 = 3.3 = {(Bv — C/ji) d x + (C f A. — Av) d y + (Afi — BX) d z }. 
(\(Cd y — Bd z ) +fi(Ad z — Cd x ) + v(Bd x — Ady)), 
which is 
= A (C'dy - B’d z ) + ya (A'd z - G'd x ) + v (B'd x - A'dy), 
where 
A' = dA — a (Bv — C/x) + h (GX — A v) + g (A ya — BX) 
= X (hG — gB) + /J, (gA — aC) + v (aB — hA ), 
with the like values for B’ and G'. Substituting the values 
(to — 1) (A, B, G) = (ax + liy + gz, hx + bg + fz, gx +fy + cz), 
we have 
(to — 1) A' = X (®g - fiz) + ya (% -23z) + v ((% - %z); 
and similarly 
(to — 1) B' = X (2\z — ®x) + ^ (fiz — %x) + v (®z — (£x), 
(in - 1) G' = X (fix - 21y) + fi (23a? - fig) + v ($x - ®y),