Contacts of Lines and Surfaces of the Second Order. 129 



forms of the two surfaces, 



V ■= x^ + o'tj + ijz + zf 

 Y=:axi/ + bz^ + ast. 

 In this case the two points of intersection of the curve of the 

 thii-d degree, and the right line on which the surfaces intersect, 

 come together, so that the right line becomes a tangent to the 

 curve. The number of conditions to be satisfied is three : there 

 is but one point of contact v/hich may be considered as the union 

 of two which have coalesced, and the species may be defined as 

 confluent-unilinear contact. 



If we throw the equations to the conoids having an unilinear 

 contact into the form 



a;{x + !/)+zt = 

 xy + z{i/ + ct)=0, 

 we obtain 



{x + ij){y + ct)-i/t = 0, 



which last equation is no longer satisfied by «=0, 2^=0, these 

 systems of roots having been made to disappear by the process 

 of elimination. 



The curve of the third degree, in which the two given conoids 

 intersect, may thus be defined as their common intersection with 

 the new conical surface defined by the third of the above equations. 



More generally, it is apparent that the three conoids, 



XU — 7/t=:0' 



yv—su=.0 

 zt—a;v = Q 



in which x, y, z, t, u, v may any of them be considered as a ho- 

 mogeneous linear function of four others, intersect in the same 

 line of the third degree. Besides which, the first and second 

 intersect in the right line y, u ; the second and third m. z, v ; 

 the third and first in x, t ; each of which lines it is evident is a 

 chord of the common cm've of intersection. For instance, y = 0, 

 ?; = may be satisfied concurrently with ail the above three equa- 

 tions by satisfying the equation zt — xv^=Q, which, as two linear 

 relations exist originally between tlie six letters, and two more 

 have been thrown in, becomes a cjuadratic equation between any 

 two of the letters. 



The only case of exception to this reasoning is, when y = 0, 

 uz=0 can be satisfied concixrrently with ~ = (), v = 0, and with 

 z = 0, / = 0; Ijut in this case the surfaces all become cones; and 

 astliere is no longer a curve of the third degree, " Cadit qufestio." 

 Even here, however, the intersection of any two of the surfaces 

 becomes a conic, and two coincident generating lines on the two 

 cones ; so that if we take one of these and tlie conic to represent 



Phil. May. S. 4. Vol. 1. No. 2. Feb. 1851. K 



