568 PROFESSOE CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CURVE. 
with the like values for (c^ . V)H and (cL . V)H. And then 
Jac. (U, H, V)H= 
A , B , C 
A' , B' C' 
P..V)H, (3„.V)H, 
in which the coefficient of A' 2 is 
=(cd,-Ba.xa > % aix, o; 
or putting for shortness 
(Cb y -Bb 2 , AB 2 -C^„ B3,-Acy=(P, Q, R); 
the coefficient is 
im, m, p<ax*> ft, o- 
52. We have 
<3=(PA + Q^-f Rv), 
and thence 
coefficient A' 2 — d$=(P$, P$2, P(§X^> v )~ Q*3, R$X^ v \ 
which is 
= /»{(ca,-Ba,)®-(aa.-cd # )a} 
+» {(ca,-BB.)«-(Ba.-aB,)a}, 
where coefficient of p is 
and coefficient of v is 
so that 
=- Aa,0-BB2l+cp£+a,£) 
= -(A3,g+B3..i + C3 2 ©)=-^ I *3,H, 
= +(A3„a+B3,»+C3,®)= 
coefficient A' ! -3Sl= 
53. And by forming in a similar manner the coefficients of the other terms, it appears 
Jac. (U, H, V)H-(3a, . . -XA', B\ C') 2 
1 
or since the determinant is 
^(A'w+3'y+Gz) 
A' , 
B' , 
G 
, 
(* » 
V 
a,H, 
bj,H, 
B,H. 
A'', B', C' 
, =0, 
a , v 
A', B\ G 
we have the required equation, 
Jac. (U, H, V)H=(B& . . -XA', B', C') 2 . 
This completes the series of formulae used in the transformations of the condition for 
the sextactic point. 
