Contacts of Lines and Surfaces of the Second Order. 127 



In the ordinary case of diploidal contact, it will be obseiTcd 

 that their latter condition does not obtain ; the four intersections 

 lie on a coincident pair of lines ; but they lie also on a crossing 

 pair, namely, in the two tangents at the points of contact. In 

 this higher species of diploidal contact, it is clear that the two 

 points of contact, which are ordinarily distinct, come together, 

 and that all four intersections coincide. 



This I call confluent contact ; the forms of U and V corre- 

 sponding thereto will be 



N=:ay'^ + aocz ] 



the common tangent at the point of contact being a? = 0, and the 

 four coincident points «'^ = 0, y^ = 0. 



The number of affirmative conditions to be satisfied being 

 three, the contact is to be entitled of the third degree. 



Observe, that it is of no use to descend below the first minors 

 in this case ; because the second minors, being minor functions 

 of X, could not have a factor in common, unless V : U becomes a 

 numerical ratio, which would imply that the conies coincided*. 



Fortified by the successful application of oui* general prin- 

 ciples to the preceding more familiar cases of contact, we are 

 now in a condition to apply with greater confidence the same 

 a priori method to the exhaustion and characterization of all the 

 varied species of contact possible between surfaces of the second 

 order ; a portion of the subject comparatively unexplored, and 

 never before thought susceptible of reduction to a systematic 

 aiTangement. 



"When there is no contact, we may write 



U=a?2 + ?/2 + 0H«^ 

 Y=ax'^ + by^ + cz'^ + dt', 



and the intersection of the suifaces will lie in each of the four 

 cones, 



{a-d)x"- + {b-d)y^+ {c-d)z^ = 



{a-c)x^+{b-c)y^-\-id-c)t^=0 



{a-b)x^+{c-b)z'' + {d-b)t'^=0 



{b-a)f+{c-a)z^+ {d-a)t^ = 0. 



Whenever the surfaces are in contact, certain of these cones 



will coincide with certain others, so that their number will be 



always less than four. Also, as we shall find in such event, 



they may degenerate into pairs of intersecting or coincident 



planes. 



* No-coiitact mill complete roincidenri" may he conceived as the two 

 extreme eases in the scale of relative conjugate forms. 



