Contacts of Lines and Surfaces of the Second Order. 131 



fonai is one, which, while it responds to the same conditions as 

 some other more general form, is incapable of being expressed as 

 a particular case of the latter, on account of the additional con- 

 dition or conditions which attach to it.^ 



I pass now to the singularities which arise from the first minor 

 determinants of U + W having a factor in common, the second 

 minors being supposed to be still without a common factor. 



"^^Tieu this common factor is linear in respect to \, let it be 

 supposed to enter not more than twice (twice, we know, by 

 the general principle enimciated at the commencement of this 

 paper, it must pass) into the complete deteniiinant. 



Two of the cones containing the intersection of U and V then 

 become coincident, and degenerate each into the same pair of 

 crossing planes. This may be termed biplanar-contact. The 

 characteristics of such contact are 



V=x^-^f + z'' + t^ 



Y = ax'^ + ai/ + bz'^ + ct^; 



the points of contact are two in number, being at the intersec- 

 tion of the two plane conies into which the ciu've of intersection 

 breaks up. The two planes in which these lie are given by the 

 equation {b — a)z^+{c — a)t'^ = 0; these intersect in the right 

 line s=0, t = 0, which meets both sm-faces in the same two 

 points, 



z=0 t = x+ \/^^?j=0 



Z = t = X-^/^lljssO; 



the two common tangent planes at these points being 



X+ V — ll/ = X— V — l2/ = 



respectively. 



This, then, is another species of double contact between two 

 conoids, and, as far as I know, the only kind hitherto recognized 

 as such. The number of conditions to be satisfied remains two, 

 as in the former species. 



Next suppose that the common factor of the first minor enters 

 three times into the complete detenninant instead of twice only, 

 as in the last case. 



The corresponding characteristics will be found to be 



V>=x^ + zt->ry^ + z^ 



V = a«2 + azt + bxf- + cz^. 



The intersection of U, V still lies ui two planes, 



but the intersection of these two planes, 



y=0 z=% 

 K2 



