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XIV. On the Sextactic Points of a Plane Curve. 
By William Spottiswoode, M.A., F.B.S., &c. 
Received June 15, — Read June 15, 1865. 
The beautiful equation given by Professor Cayley (Proceedings of the Royal Society, 
vol. xiii. p. 553) for determining the sextactic points of a plane curve, and deduced, as 
I understand, by the method of his memoir “ On the Conic of Five-pointic Contact ” 
(Philosophical Transactions, vol. cxlix. p. 371), led me to inquire how far the formulae 
of my own memoir “ On the Contact of Curves ” (Philosophical Transactions, vol. clvii. 
p. 41) were applicable to the present problem. 
The formulae in question are briefly as follows : If U=0 be the equation of the curve, 
H=0 that of its Hessian, and V =(a, b, c,f g, h)(x, y, zf= 0 that of the conic of 
five-pointic contact ; and if, moreover, a, /3, y being arbitrary constants, 
b=ux-\-fiy-\-yz, 
□ = (y^U - U)d, + («B,U -- yBJJ)^ + (,3d,U - «d,U)b 2 , J ' 
then, writing as usual 
BJJ=w, bJJ=w; ^H=^, B,H=r, 
^i=v l w 1 —u' 2 , . . Jf=v'w'—u 1 u', . . 
vy — w(3=X, wot — uy—gj, u\ 3 — vu—», 
the values of the ratios a : b : c :f : g : h are determined by the equations 
v=o, □ v=o, □ 2 v=o, □ 3 v=o, n 4 v=o. . . . 
Now, if at the point in question the curvature of U be such that a sixth consecutive 
point lies on the conic V, the point is called a sextactic point ; and the condition for this 
will be (in terms of the above formulse) □ 5 Y=0. From the six equations Y=0, 
□ Y=0, . . D 5 Y=0, the quantities a , b, c, f, g , h can be linearly eliminated; and the 
result will be an equation which, when combined with U = 0, will determine the ratios 
x:y:z, the coordinates of the sextactic points of U. But the equation so derived con- 
tains (beside other extraneous factors) the indeterminate quantities a, (3, y, to the 
degree 15, which consequently remain to be eliminated. Instead therefore of pro- 
ceeding as above, I eliminate a, (3, y beforehand, in such a way that (W=0 repre- 
senting any one of the series Y=0, □V=0, . . from which a, (3, y have been already 
mdccclxv. 4 x 
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