136 Mr. J.J. Sylvester's Enumeration of the 



In this case the common enveloping cone becomes identical with 

 the plane (considered as a coincident pair of planes) in which 

 the svu'faces intersect. 



The characteristics take the form 



U = .r'^ + xi/ + zt 

 Y = xy -I- zt. 



The intersection is contained completely in the common tan- 

 gent plane ,r = Oj and consists of the two right lines, 

 (a'=0 r = 0) 

 (j? = ^ = 0). 



This, the highest and crowning species of contact, I call bili- 

 ueo-indctinite. It is defined by six conditions. 



At each point of the two lines of intersection of U and V there 

 is contact, and a \'ery ])eculiar species of contact at the intersec- 

 tion of these two lines themselves. 



To form a distinct idea of this, let the physical visible or 

 quasi-visible intersection of U, V take place along the two lines 

 L, M ; the rational intersection must be conceived as made up 

 of the wry quadrilateral, L, ]M ; L', M', in which L is indefi- 

 nitely near to L', and J\I to ]\1'. It follows, therefore, that there 

 is contact at the four angles of the quadrilateral ; but as there is 

 nothing to fix the relative directions of the diagonal joining the 

 intersection of L and ]\I to that of L' and i\I', because there is 

 nothing to restrict the position of tbe latter point, except that 

 it shall lie upon either surface*, it appears that not only is there 

 contact at the junction of the two lines constituting the complete 

 intersection of the two surfaces, but that these surfaces continiie 

 to touch at consecutive points taken all round this first, and in- 

 definitely near to it in any direction t- 



Bilineo-indefinite (the highest) contact for two conoids is 

 strictly analogous to confluence, the highest species of" contact 

 between conies. For this latter may be conceived as an inter- 

 section made up of two coincident pairs of coincident points ; 

 and the former, as an intersection made up of two coincident 

 pairs of crossing right lines ; and a pair of crossing lines is to a 

 plane locus of the second degree what a coincident pair of points 

 is to a rectilinear locus of the same degree. 



In the subjoined table I have brought under one point of view 

 the characters and algebraic fonns which I call the condensed 

 forms corres))onding to each species of contact above detailed. 



* Tliis will be better seen by reference to the analogj- presented by the 

 case when the two conoids touch all along a curve. The rational intersec- 

 tion is made up of this curve and another indefinitely near it. The two 

 curves, whatever be the ])osition of their node, will lie in the same enve- 

 loping cone, so that the jjosition of the node is indeterminate. 



t As the two surfaces jut one close into the othci* at this point, it would 

 perhaps be not improper to designate the contact at such jioiut as unibilical. 



