222 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
and Jac. denotes the Jacobian or functional determinant, viz. 
Jac. {U, H, ¥) = 
9 X U, 
9 yJJ, 
9 Z U 
9 Æ 
d y H, 
9 Z H 
9^, 
9,¥ 
and Jac. (U, H, 12) would of course denote the like derivative of (U, H, 12); the 
subscripts ( H , p) of 12 denote restrictions in regard to the differentiation of this 
function, viz. treating 12 as a function of U and H, 
H = (2(, SB, ®, & @, V, c', /', 2/', 2/, 2/0 
if (a', 6', c'y g', K) are the second differential coefficients of H, then we have 
9*12 = (9*21,..£ a,..) (=a*ao 
+ ( 21,..$9*«',..) (=9*00; 
viz. in 9*12# we consider as exempt from differentiation (a!, b', c', f, g', h') which 
depend upon H, and in 9*0# we consider as exempt from differentiation (21, 23, (S, %, @, ,£)) 
which depend upon /7. We have similarly 
d y £l = d y £lji + d y £lv, and d z Q = 9*12#+ 9*12#; 
and in like manner 
Jac. {JJ, H, 12)= Jac. {JJ, H, 12#) + Jac. (U, 77, 12#), 
which explains the signification of the notations Jac. (JJ, H, 120, J ac - (U, H, 12p). 
The condition for a sextactic point is in the first instance obtained in a form 
involving the arbitrary coefficients (A, g, v); viz. we have an equation of the order 5 
in (A, g, v) and of the order 12m — 22 in the coordinates {x, y, z). But writing 
= \x + gy + vz, by successive transformations we throw out the factors ^ 2 , 
thus arriving at a result independent of (X, g, v); viz. this is the before-mentioned 
equation of the order 12m —27. The difficulty of the investigation consists in obtaining 
the transformations by means of which the equation in its original form is thus 
divested of these irrelevant factors. 
Articles Nos. 1 to 6.—Investigation of the Condition for a Sextactic Point. 
1. Following the course of investigation in my former memoir, I take (X, Y, Z) 
as current coordinates, and I write 
T = (*J[X, Y, Z) m =0 
for the equation of the given curve; (x, y, z) are the coordinates of a particular 
point on the given curve, viz. the sextactic point; and JJ, = (*$#, y, z) m , is what T 
becomes when (x, y, z) are written in place of (X, Y, Z): we have thus /7 = 0 as a 
condition satisfied by the coordinates of the point in question.