246 ON THE SEXTACTIC POINTS OF A PLANE CURVE. [341 
Article Nos. 51 to 53.—Proof of identity used in the fourth transformation, viz. 
Jac.(U, V, H)H=-EV, or say Jac. {U, H, V)tf = (02l, B', CJ. 
51. We have 
V = ((Si, @,£X, ft v), (§, 8, ft, v), (@, g, ft v)ldz, d„ a,), 
and thence 
0*- V =((9*21, 0*4?, 0z@$X, y, V), (0*4?, 0«;23, 0*5$X, y, V), (0*@, d x %, d x (f$\, fji, v)Jd x , dy, d z ), 
and 
(0*. V)H = ((0*21, 0*4?, 0*®][X, /a, v), (03,4?, 0*23, 0*5#X, /A, v), (0*@, 0*5, 0*@#X, /a, v)\A\B', C'), 
with the like values for (d y . V) H and (d z . V) H. And then 
Jac. (17, H, V)iT = 
A , B , C 
A' , B' , O' 
(0*.V)i7, (d y .V)H, (0 2 . V)H, 
in which the coefficient of A’ 2 is 
= (Cdy-Bd z ) (21, £, @$X, /A, *); 
or putting for shortness 
(Cd y -Bd 2 , A02-<70*, P0*-A0j,) = (P, Q, A); 
the coefficient is 
52. We have 
(P21, P4?, P©#X, /A, *,). 
0 = (PX + Qy + ifo), 
and thence 
coefficient A' 2 — 021 = (P21, P4?, P@£x, /a, *) - (P2(, Q2i, P21&X, /a, *) 
= ft 81) 
+ V {(Cdy - Bd z ) <8-(Bdz- 5IS„) aj, 
where coefficient of fi is 
= - Ad£\ — Bd z $ + C (0*21 + 01,4?) 
and coefficient of v is 
= - (A0*2[ + Bd£ + G®*®) = - ^-=-- xd z H, 
= + (Ad y 9t + Bdy$ + Cdy®)= ^ x xdyll, 
so that 
coefficient A' 2 — 021 = — x (ud,H — vd v H). 
to — 1