132 Mr. J. J. Sylvestei^'s Enumeration of the 



meets the surfaces in the two coincicleut points, 

 y=0 c = a:- = 0. 



This, therefore, I call confluent-biplanar contact; the two 

 conies constituting the complete intersection, instead of cutting, 

 touch and at their point of contact the two conoids have a con- 

 tact of a superior order. The conditions to be satisfied for this 

 ease are three in number. 



Next suppose that the common factor of the first minors enters 

 only twice into the complete determinant, but that the remaining 

 two factors became equal. 



Here the analytical characters of unilinear and biplanar con- 

 tact are blended ; in fact, the intersection consists of a conic and 

 a pair of right lines meeting one another and the conic. The 

 characteristics are 



'[]=x'^ + y'^ + z'^ + zt 



The intersection is contained in the two planes 

 z=Q[b-a)z^{c-a)t = 0, 

 and consists of the two lines ^• = 0, 5r^ + ?/- = 0, lying in the com- 

 mon tangent plane z=0, and the conic 



[h~a)z—{c—a)t = Q \ 



{a-c)x''-^{a-c)i/Jr{b-c)z'^ = QS 



There are three points of contact, viz. the point ^"=0, ^^=0, 

 ^=0, where the two right lines cut, and x^ + y^ — O, t=0, z=0, 

 where these lines meet the conic. This, then, is a case of triple 

 contact. I distinguish it by the name of bilinear-contact. The 

 number of conditions is still three. 



Now all else remaining as before, let the two pairs of equal 

 roots in the complete determinant become identical, or, in other 

 words, let the common factor of the first minors be continued 

 four times in the complete determinant. The characteristics 

 become 



lJ = xz + xt + y'^ + z- 



y = axz + bxt + by- + bz^. 

 The intersection becomes the two right lines 



.r = 2/^-f.^2=0, 

 and the conic 



^=0 x''- + f-=Q). 

 All these meet in the same point, 



^ = y = 2r=0; 



so that instead of contact in three points, the contact takes place 

 about one only, in which the three may be conceived as merging. 





