124 Mr. J. J. Sylvester's Enumeration of the 



Two of the pairs of lines become identical, i. e. two of the four 

 points of intersection coincide. 



This may he termed " simple contact." The tangent at the 

 point of contact is x = 0; this equation making U and V each 

 become of only one order. 



The intersections are 



a;=0 ij=0 (1.) 



a:=0 y = (2.) 



\^a^.x+ \/6-c.y=0...5=0. . . (3.) 



\^a—c,x— \^b — c.y = 0...s=0. . . (4.) 



These are obtained by making V — «U = 0, which gives x=0 or 

 z=0. 



07=0 gives y^=0, i. e. y = twice over, and s=0 gives 



{x-c)x''-\-{b-c)y^=0. 



The number of conditions to be satisfied in this case is one only. 

 Next let I I (U + \V) have all its roots equal. This condition 

 will be satisfied (still leaving U and V as general as they can 

 remain consistent \nth these conditions) by making 



\]ssx^-\-ys + yx 



V = ax'^ + ayz + hyx. 



tions in tlielr present forms, contain an arbitrariness of 10 degrees j viz. 9 on 

 account of x, y, z being arbitrary linears of f, j?, ^ ; 2 on account of the ratios 

 a:h:c; together 1 1 reduced by one degree on account of a?, y, z, changed into 

 Ix, ly, Iz, leavingU=0, V=0 unaiFected. Now the degrees of arbitrariness in 

 two conies, subject to satisfy only one condition, is 2 X 5 — 1 or 9. Hence there 

 is one degree of arbitrariness to spare. In fact, on making a-=.b, the axis z 

 becomes the line joining the two points of intersection distinct from the 

 point of contact ; x remaining the tangent at the point of contact, and y, 

 strange to say, still arbitrary, subject only to passing through the point of 

 contact ; if, however, y be made to pass thi'ough the point of contact, and 

 either one of the distinct intersections, this form, 



\=:iax''-\-ay'^+cxz, 

 becomes no longer tenable, but gives place to 

 \]-=.y^-YyX-\-XZ 

 \r=.ay^-\- ayx + cxz, 



where x is the tangent at the point of contact, k the line joining the two 

 intersections with one another, and x, x+y respectively the hues joining 

 either of them with the point of contact ; if the multiplier of yx in V in the 

 above be made b instead of a, {x) remains the tangent as before, y becomes 

 any hne through the point of contact, and z any line through one of the 

 distinct intersections. A systematic view of similar modulations of form 

 and the study of the laws of arbitrariness connected with them, as apph- 

 cable to the general subject-matter of this paper, must be deferred to a sub- 

 sequent occasion. 



