134 Mr. J. J. Sylvestei-'s Enumeration of the 



anotlier, but still not coinciding in the same planes ; so that the 

 actual visible or quasi-visible* intersection wvW be in three right 

 lines, of which the middle one meets each of the two others. 



This will further appear by proceeding regularly to solve the 

 equations 



U = V = 0. 



V— cU = gives y= ±K,r, where K = \ / - — ^ and therefore 



V c—b 



xs-\-kxt = 0, or xz — kxt = 0; whence we see that the complete 



intersection is represented by the lines 



{a?=0 y = 0); {z + kt=Q y-kx=0) 



(a?=0 y = 0); \s-kt = Q y-\-kx=Q), 



showing that there are but three physically distinct lines, as 



already premised. 



This, then, may be considered as derived from the preceding 

 case of a wry quadi'ilateral intersection, by conceiving two oppo- 

 site sides of the quadrilateral to come indefinitely near, but 

 without coinciding. 



Let these two lines be called P and P' ; take any point in P 

 and any two points in P' indefinitely near to one another and the 

 point first taken, then this indefinitely small plane will be com- 

 mon to both surfaces, and consequently they ought to touch 

 along every point in the line P. This is again confirmed by the 

 forms given to U and V. For at any point where the coordi- 

 nates are 0, 0, f, 6 the equations to the tangent planes to the 

 two surfaces respectively are 



f^ + % = 

 c^x-{-cdy=0, 

 that is to say, are identical. 



^Tiilst, therefore, certain grounds of geometrical, and still 

 stronger grounds of analytical analogy, might seem to justify 

 this species of contact taking the name of confluent quadiilinear, 

 yet as, in fact, tlie intersection is trilinear, and as, moreover, the 

 two indefinitely proximate lines must be considered, not as coin- 

 cident, but as turned away from one another through an indefi- 

 nitely small angle and out of the same plane, I prefer to take 

 advantage of this striking property of contact at every jjoint along 

 a line (a property entirely distinct from any that we have yet 

 considei'ed), and confer upon the species of contact we have been 

 considering the designation of unilinear-indefinite contact. 



A^Tiere the line of indefinite contact meets the two other lines 

 of the intersection, the contact is of course of a higher order ; 

 thus offering a parallel to what takes place in ordinary unilinear 



* I use the term quasi-visible, because the iutersection may become in 

 part or whole imagiiiaiy. 



