552 PROFESSOR CAYLEY. ON THE SEXTACTIC POINTS OE A PLANE CURVE. 
15. Hence the condition will contain the factor 9, and throwing out this, and also the 
constant factor 1 . it becomes 
m — 2 
(_ 15 w »+60m— 57)HJac.(U, V , H)H 
+(30m-54)(m-2) HJac.(U, VH, H) 
+(m-2) 2 (9H 2 BQ-45HQBH+40^H)=0. 
16. We have 
B J (VH)=p..V)H+B # VH, 
viz. in (B,*. V)H, treating V as a function of ( x , y, z ) we operate upon it with B* to 
obtain the new symbol B* . V, and with this we operate on H ; in B it V we simply mul- 
tiply together the symbols B*. and V, giving a new symbol of the form (B 2 , B^, B a B„) 
which then operates on H. We have the like values of B y (VH) and B 2 (VH); and 
thence also 
Jac. (U, VH, H)= Jac. (U, V, H)H+ Jac. (U, VH, H), 
viz. in the determinant Jac. (U, V, H) the second line corresponding to V is B*. V, 
Bj, . V, B s . V (V being the operand ) ; and the Jacobian thus obtained is a symbol which 
operates on H giving Jac. (U, V, H)H ; and in the determinant Jac. (U, VH, H) the 
second line is B, r VH, B y VH, B S VH (V being simply multiplied by B*, B^, B. respectively). 
17. Substituting, the condition becomes 
(— 15m 2 + 6 0m— 57) HJac.(U, V, H)H 
-f(30m-54)(m-2){H Jac. (U, V, H)H+ Jac. (U, VH, H)} 
+ (m—2) 2 {9H 2 BO-54HOBH + 40^BH}=0, 
or, what is the same thing, 
(15m 2 -54m+51)H Jac. (U, V , H)H 
-f (30m-54)(m-2)H Jac. (U, VH, H) 
+(m-2) 2 {9H 2 BO-45HOBH-f40^BH} = 0. 
Article Nos. 18 to 27. — Fourth transformation, and final form of the condition fora 
Sextactic Point. 
18. I write 
(5m-12)QBH-(3m-6)HBG=9- Jac. (U, 12, H) (J) 
OBH+ HB12= B(QH), 
and, introducing for convenience the new symbol W, 
-50BII+ HBO=W, 
5m- 12, —(3m— 6), 9 Jac. (U, O, H) 
1 , 1 , B.12H 
-5 , 1 , W 
= 0 , 
so that 
