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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 (Phil. Trans 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 α, β, γ, δ, β', γ', δ' I consider the come of five-pointic contact of a section of the surface made by the planekϖ' =0, where ϖ = αy + βy + γz + δt, and ϖ' = α'y + β'y + γ'z + δ't, andk is indeterminate; and then proceed to determinek , and thereby the azimuth of the plane about the lineϖ = 0,ϖ' = 0, so that the contact may besixpointic. The formulæ 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 inϖ :ϖ =k , and besides this of degree 12n —27 in the coordinatesx, y, z, t (n being the degree of the surface), giving rise to the theorem above stated.

IV. On the contact of conics with surfaces