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



a degenerate form of a line of the third degree, the remaining 



straight line passes through a double point of such degenerate 



form, and the case passes into that of confluent unilinear contact. 



The two double points in the intersection of the two conoids 



\] = x[x + y)-\-zt=Q 



Y=xy-^z{y + ct)=0, 



by which I mean the points of intersection of the conic with the 



right line common to them, is found by making a- = 0, z = 0, 



and substituting in the derived equation 



which gives ?/ = 0, or y+ (c — 1)^ = 0; so that the two points 

 required are 



^ = y = r = 



^=0 y={l—c)t z = Q. 

 It appears also that the entire intersection is contained in each 

 of the two cones, 



U-V, i. e. ^2+ r((l_c)/-y) 

 and 



cU— V, i. e. cx^ + y(jc — \)x — z'^, 



the respective vertices of which are at the points above deter- 

 mined. 



The equations for confluent unilinear contact, 



x{x—y)+z{y + t)—0 

 xy + z{cz + t)=Q, 

 give 



{x-Vy){cz + t)-{y + t)y=0; 



which, on making x=Q, z=0, is satisfied by y^=0; showing 

 that the confluence takes place at the point 



,r=0 y = z=0. 



The number of terms in the two equations for ordinary uni- 

 linear contact being six, and in those given for confluent unili- 

 nears seven, and the empirical rule in all other cases being that 

 the terms tend to diminish and never increase in number as the 

 degree of the contact (expressed by the number of conditions to 

 be satisfied) rises, I am led to suspect that the conjugate system 

 for the latter species of contact may admit of being reduced to 

 some more simple form. 



I must state here once for all, that all the distinct systems of 

 (at least consecutive) conjugate forms that have been, and will 

 be given, are mutually untransformable. This it is which di- 

 stinguishes singular from, particular forms. 



A particular form is included in,its primitive ; but a singular 



