250 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
and we then obtain the equation of the contact-locus in the form 
(ae — 4bd + 3c 2 ) 3 — 27 (ace + 2bed — ad 2 — №e — c 3 ) 2 = 0, 
which is a onefold locus of the order 6. It follows that we have 
(1/cl, !.-.)= 6, agreeing with (1/cl, 1 ,\) = n 4- 2m — 3. 
The condition in order that the conic may touch a given line is given by an 
equation of the form 
(*\a 2 , ab, b 2 , 2ce — 3d 2 , ae — 8bd, ad-126c) 1= =0, 
which is a onefold locus of the order 2; it at once follows that we have 
(1/cl, 1 :/) = 12, agreeing with (1/cl, 1 :/)= 2n 4- 4m — 6. 
It is a matter of some difficulty to show that we have 
(1/cl, 1 • //) = 18, agreeing with (1/cl, 1 • //) = 4>n + 4m — 6 ; 
but I proceed to effect this, first remarking that I do not attempt to prove the 
remaining case 
(1/cl, 1///) = 15, agreeing with (1/cl, 1 ///) = 4m 4- 2m — 3. 
Investigation of the value (1/cl, 1 • //) = 18 : 
We have the sextic locus 
(ae — 4bd + 3c 2 ) 3 — 27 (ace 4- 2bed — ad 2 — b 2 e — c 3 ) 2 = 0, 
and combined therewith two quadric loci, 
(* ][a 2 , ab, 6 2 , 2ce — 3d 2 , ae — 8bd, ad — 12be) 1 = 0, 
(*'$n 2 , ab, b 2 , 2ce — 3d 2 , ae — 8bd, ad — 126c) 1 = 0, 
which intersect in a threefold locus of the order 24; it is to be shown that this 
contains as part of itself the quadric threefold locus (a = 0, 6 = 0, 2ce — 3d 2 = 0) taken 
three times, leaving a residual locus of the order 24 — 6, =18. 
We may imagine the coordinates a, b, c, d, e expressed as linear functions of any 
four coordinates, and so reduce the problem from a problem in 4-dimensional space to 
one in ordinary 3-dimensional space. We have thus a sextic surface, and two quadric 
surfaces; the sextic is a developable surface or torse, having for one of its generating 
lines the line a — 0, 6 = 0, and for the tangent plane along this line the plane a = 0; 
the two quadric surfaces meet in a quartic curve passing through the two points 
(a = 0, 6 = 0, 3ce — 2d 2 = 0), which are points on the torse; it is to be shown that each 
of these points counts three times among the intersections of the torse with the quartic 
curve, the number of the remaining intersections being therefore 24 — 6, = 18 ; and in order 
thereto it is to be shown that each of the points in question (a = 0, 6 = 0, 3ce — 2d 2 = 0)