MR, W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SURFACES. 305 
which is the same as if A, B, C, D had not been affected by the differentiations or by 
the operation A. The expression is therefore a function of x, y, z, t, A, B, C, D, and 
not of a, (3, . . , a', j3', . ., excepting so far as they appear in A, B, C, 1 ). If, therefore, 
we put <p=wX — aP, <p,=wY — yP, <p. 2 =u r /j — zP, we conclude that (36), when divested of 
its extraneous factors (say, O), is a function explicitly of the degree 12n — 27 in x , y, z, t, 
and of the degree 10 in A, B, C, D. But A, B, C, D are themselves linear homogeneous 
functions of ra-, N ; so that the expression in question may be regarded as explicitly of 
the degree 12^—27 in x, y , z, t, and of the degree 10 in N. This equation, solved 
for k'—zff : N, will consequently give ten positions of the cutting plane all passing through 
the point ( x , y, z, t ), for which the curve of section is sextactic at the point. Hence the 
theorem, “ If L be any line through any point P of a surface, ten conics may be drawn 
in planes passing through L having six-pointic contact with the surface at the point.” 
It is to be observed that the plane of section is not necessarily normal ; but if the line 
whose six coordinates are («, b, c, f, g, h) be made to coincide with the normal at the 
point whose rectangular coordinates are x, y , 2 , it will follow that 
a : b : c=u : v : w 
Aw-j-By-j-Cw=0 
fu -f -gv -\-Jm =0, 
with other relations which would abbreviate, although not essentially simplify, some 
previous expressions. The results of placing the line in the tangent plane are noticed 
below. 
§ 5. Note on tangents of more than t wo-pointic contact. 
If V, instead of being as hitherto quadric, be linear, we shall have the case of tangents 
to the curve of section of U with the plane 7tJ 1 lb 7JS 0. And if the contact be three- 
pointic, each of the ratios d x V : u, : v, „ . will be, in virtue of (20) of § 2, equal to 
AV : H. But since V is linear AV=0, and consequently the condition for a tangent of 
three-pointic contact is 11 = 0. Now 
H=(a, • .)(A, B, C, H) 2 
= (£l, . .)(a , OT — aN, (3 'gt — [BN, yW — yur ' , h'vr — ^cf) 2 
= w ' 2 (£l, . .)(«'#-«, (3'hf — j3, y'hf — y, h'h J -b) 2 ; 
so that the condition H = 0 may be written 
(a, . .)« /3 1 , y, s'/v*-2(a, . .)(«', (3', y, s’x«, r , ^+(a, . .)(«, (3, r, . (4i> 
which will determine two values of Jc, and consequently two positions of the cutting- 
plane for which the tangent line has three-pointic contact. 
It may be noticed that in the solution of the equation above written there occurs the 
following expression : 
(a, . .)(«', i3 \ y, ij . (a, . .)(<*, (3, r , s)*— [(a, . .)(«', P, »x«. P, r. >)?. 
MDCCCLSX. 2 S 
