132 
PROFESSOR CAYLEY ON THE CURVES 
It follows that the number of the conics (1 : :) is = 9, which agrees with the general 
value (1: :) = 2 If the conic pass through the cusp we have c= 0, and the equation 
in b is reduced to a quartic ; it is convenient to alter the letters in such wise that the 
quartic equation may be obtained in the standard form (a, b, c, d, e$b, 1) 4 =0 ; viz. this 
will be the case if the equation of the conic is taken to be 
(c, 6c, 0, \a, 2b, 2 djx, y, zf= 0, 
and we then obtain the equation of the contact-locus in the form 
(ae-m+3cy-2 7 (ace +2bcd-ad 2 -b 2 e- cj— 0, 
which is a onefold locus of the order 6. It follows that we have 
(1*1, 1 .\)=6, agreeing with (1*1, 1 .\)=n-\-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 , 2cc— 3<f, ae—8bd, ad—12bc) 1 — ^, 
which is a onefold locus of the order 2 ; it at once follows that we have 
(1*1, 1, :/)=12, agreeing with (1*1, 1, :/)—2n-\-A.m— 6. 
It is a matter of some difficulty to show that we have 
(1*1, 1, • //) = 18, agreeing with (1*1, 1 • //)=4w+4m— 6; 
but I proceed to effect this, first remarking that I do not attempt to prove the remaining 
case 
(1*1, 1 ///)=15, agreeing with (1*1, 1 ///)=4%+2m— 3. 
Investigation of the value (1*1, 1 • //) = 18: 
We have the sextic locus 
(ae—4:bd -f- 3c 2 ) 3 — 27 (ace -f 2 bed— ad 2 — b 2 e—c z ) 2 = 0, 
and combined therewith two quadric loci, 
( 2 , ab, b 2 , 2ce—2>d 2 , ae—8bd , ad— 125c) 1 = 0, 
(#'X® 2 , ab, b 2 , 2ce—Sd 2 , ae—8bd, ad—\2bc) 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 («=0, b=0, 2ce—3d 2 =0) taken three times, 
leaving a residual threefold 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, b—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, b= 0, 
3cc — 2d 2 =Q), which are points on the torse ; it is to be shown that each of these points 
