Contacts of Lines and Surfaces of the Second Order. 133 



Tliis I call conflueut-bilinear contact. It requires the satisfac- 

 tion of four conditions. 



Next let lis suppose that the two distinct factors arc connnon 

 to each of the first minors. This will imply the existence of 

 fom* affirmative conditions. 



The complete determinant will of necessity contain each of 

 these factors twice, so that no additional singularity can enter 

 through this determinant. The characteristics assume the form 



V = X^ + i/ + Z^+t^ 



V = a.v- + ai/ + b;:^ + bt\ 



The two surfaces will meet in four straight lines, forming a wry 

 quadrilateral, whose equations are 



2± \/'-it=o. 



These intersect each other in the four points 

 ^=0 y = -^ + t^ = 

 -=0 t = w'^ + f = 0, 

 each of which will be a distinct point. Tliis I term quadi'iliuear 

 contact. 



Now let the two factors common to each of the first minors 

 become identical ; so that a squared function, instead of an ordi- 

 nary quadratic function of \, is now their common measure. 



The factor which enters twice into each of the first minors 

 mil enter four times into the complete determinant ; the number 

 of conditions to be satisfied is one more than in the preceding 

 case, namely five, and the characteristics become 

 \] = a^--{-y^ + X'z + yt 

 V = a.v^ + by- + cxz + cyt. 



Here arises a singularity of form in the intersections utterly 

 unlike anything which has been remarked in the preceding cases. 

 For it \^'ill not fail to have been obsei-ved, that the intersection 

 in the nine preceding cases was always a line or system of lines 

 of the fourth degree, so as to be cut by any plane in four points. 



But in this case, the fact of the first minor having a factor in 

 connnon, shows that the intersection is contained in two planes 

 (which is of course to be viewed as a degenerate s])ecies of cone) ; 

 and the fact of the complete determinant having all its roots 

 equal, shows that there is but one system of a pair of planes in 

 which the intersection is contained, and no more. 



So that tlic two pairs of planes, into which the wry quadrila- 

 teral was divisible in the case immediately ])recediHg, now ));'Come 

 a single pair. This can only be explained by two of the op])osite 

 sides of tlie quadrilateral becoming indefinitely near to one 



