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XVI. On the Contact of Conics with Surfaces. 
By William Spottiswoode, M.A., F.B.S. 
Received February 16, — Read March 10, 1870. 
It is well known that at every point of a surface two tangents, called principal tangents, 
may be drawn having three-pointic contact with the surface, i. e. having an intimacy 
exceeding by one degree that generally enjoyed by a straight line and a surface. The 
object of the present paper is to establish the corresponding theorem respecting tangent 
conics, viz. that “at every point of a surface ten conics may be drawn having six-pointic 
contact with the surface ; ” these may be called Principal Tangent Conics. In this 
investigation I have adopted a method analogous to that employed in my paper “ On 
the Sextactic Points of a Plane Curve” (Philosophical Transactions, vol. civ. p. 653); 
and as I there, in the case of three variables, introduced a set of three arbitrary constants 
in order to comprise a group of expressions in a single formula, so here, in the case of 
four variables, I introduce with the same view two sets of four arbitrary constants. If 
these constants be represented by «, (3, y, ^ ; a', /3', y', h', I consider the conic of five- 
pointic contact of a section of the surface made by the plane w — AW = 0, where 
?z=u,x-\-fiy- : \-yz J r lt, and zz'=a'x-\ -|S , ^ + y '2 + ^, and k is indeterminate ; and then I 
proceed to determine k, and thereby the azimuth of the plane about the line ^ = 0, 
ro-' = 0, so that the contact may be six-pointic. The formulae thence arising turn out to 
be strictly analogous to those belonging to the case of three variables, except that the 
arbitrary quantities cannot in general be divided out from the final expression. In fact, 
it is the presence of these quantities which enables us to determine the position of the 
plane of section, and the equation whereby this is effected proves to be of the degree 
10 in : Tz'=k, and besides this of the degree 12% — 27 in the coordinates x, y, z, t, 
giving rise to the theorem above stated. 
Beyond the question of the principal tangents, it has been shown by Clebscii and 
Salmon that on every surface U a curve may be drawn, at every point of which one of 
the principal tangents will have a four-pointic contact. And if n be the degree of U, 
that of the surface S intersecting U in the curve in question will be 11% — 24. Further, 
it has been shown that at a finite number of points the contact will be five-pointic. The 
number of these points has not yet been completely determined ; but Clebscii has shown 
(Crelle, vol. lviii. p. 93) that it does not exceed %(11% — 24) (14%— 30). Similarly it 
appears that on every surface a curve may be drawn, at every point of which one of the 
principal tangent conics has a seven-pointic contact, and that at a finite number of points 
the contact will become eight-pointic. But into the discussion of these latter problems 
I do not propose to enter in the present communication. 
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