341] 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
239 
Article Nos. 36 to 40.—Proof of equation (0.V) H= Jac. (U, H, Ф), 
used in the second transformation. 
36. We have 
V =(5i, fi, v~$d x , d y , d z ) 
= № + % + ©0„ $д х + Щ + %д г , @0 я + ^+едх, fi, v). 
Also 
Э = (Bv — Cfjb) д х + (C'A — Av) д у + (Afi — ВХ) d z 
== ХР 4- fiQ -f- vR, 
if for a moment P, Q, R = Gd y -Bd z , Ad z -Cd x> Bd x -Ad y . 
Hence 
0 . V = (PX + Qfi + Rv) . (210* + ЬЬу + ®0 г , £0* + 530, + ЪЬг, ©Э* + %д у + (ВДА, fi, v), 
viz. coefficient of A 2 
= pm x +p$d y + p®d z , 
and so for the other terms ; whence also in (0. V ) H the coefficients of X 2 , &c. are 
(рт х +р$д у +р®д г )н, &c. 
37. Again, in Jac. ( U, H, 4>), where <f> = (21, 23, (£, $, @, >£)$A, ¡i, v) 2 , the coefficients 
of A, 2 , &c. are Jac. (U, H, 2(), &c. ; and hence the assumed equation 
(0 . V ) H = Jac. ( U, H, Ф), 
in regard to the term iu A 2 , is 
(P210* + P$d y + Р@0 г ) H = Jac. ( U, H, 21), 
and we have 
Jac. (U, H, 21) = 
A , B , G 
д х Н, д у Н, d z H 
21 
= [д х Н (Сд у - Вд г ) + д у Н (Ад г - Gd x ) + д г Н (Вд х - Аду)] 21, 
= (д х Н. Р + д у Н. Q + д г н. R) 21 ; 
so that the equation is 
P№ X H + PftdyH + P®d z H, 
= P№ X H + QAd y H + RW Z H, 
or, as this may be written, 
[(50, - Gd y ) $ - (Gd x - Ad z ) 21] d y H 
+ [(Bd z - Gdy) © - (Аду - Bd x ) 21] d z H = 0.