564 PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
We thus obtain 
+(«, . .Jx, y, z) 2 (9f, . .Jda, . .) 
— 2(9f, • .Jax+Tiy+gz, . .Xxba+ifbb+z^c , . .) 
{a,..Jx,y, Jo,..) 
-(d£T, . -Xax+hy+gz, . .) 2 
2 (a , . Jjr, y, zjdx, Sy, Sz)(9P, . •!«, . •) 
.Xa%+hy+gz, . XaBar+ASy+ySs, .) 
= (*,.. X*,y, 
— (hgl, . .Xa'x+h'y+g'z, . .) 2 , 
= (da',..X*, y, *)’(&.. X«', ..) 
+(®', . -X#, y, ^) 2 (9[, . -X^a', . .) 
— 2(91, . .Xa'x+h'y+g'z, . .Xxda'+ybti+z'dg', . 
=2(a', . .Xx, y, zXdx, Sy, S*)(9k . -X«', • •) 
-2(91, .Xdx+Ky+tfz , . •Xaftff+A'Sy+y'Sas, 
45. If in these equations respectively we suppose as before that («, b, c,f, g, h) are the 
second differential coefficients of a function U of the order m, and (a-, b 1 . c',f , g', hi) 
the second differential coefficients of a function U 7 of the order m'; and that (A, B, C), 
(A', B', C') are the first differential coefficients of these functions respectively, then after 
some easy reductions we have 
(m-l)(m-2)SU(9f, . Ja , . .) = . .X</, . .) 
+m(m— l)U(9f, . Jba, • •) — (m'— 1) 2 (S9L, . ^A', B', C') 2 , 
— 2(m— l)(m— 2)(91', . ^A, B, CXSA, SB, SC) 
m(m- 1)U(B9T, . .X«'» • •) = (m'-l)(m , -2)SU , (91, . .X«', . .) 
— (m— 1) 2 (S9P, . .XA, B, C) 2 +m'(m'-l)U'(9, • -X^< • •) 
— 2 (m' — l)(m' — 2) (91, . .XA', B', C'XSA', SB', SC'). 
2(m-l)SU(91', . .Xa, . .) = 2(m'-l)SU'(& . .Jet, . .) 
-2(m-l)(9f, . -XA, B, CXBA, SB, SC) -2(m'-l)(91, . -XA', B', C'X^A', SB, SC'), 
equations which may be verified by remarking that their sum is 
m(m— l){SU(9f, . -X®, • 0+U[(Sl', . OP®, . O+PS', • 0C«> • •)]} 
— (m — l) a {S9L', • -XA, B, C) 2 +(&', . .XA, B, C^A, SB, SC)}=m'(m'-l) &c., 
viz., this is the derivative with S of the equation 
m(m— 1)U(9P, . .X«, • . )-(m-l) 2 (! 9', . -XA, B, C) 2 =m'(m'-1) &c. 
46. Taking now U'=H, and therefore m'=3m— 6 ; putting also U=0, SU=0, and 
writing as before 
E^B =(S91, . -XA', B', C') 2 , 
FT r =(9(, . . XA', B', C'X^A', SB', SC'), 
E¥‘=(Sa , J ..XA, B, C) 2 , 
?%=(%', • . XA, B, CX^A, SB, SC), 
EH =(S3, ..x«'> ••)> 
m=(9, .. x^®'* • •)> 
