WHICH SATISFY GIVEN CONDITIONS. 
133 
count 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, 5 = 0, 3 ce — 2(2 2 =0)is situate 
on the nodal line of the torse, and that the quartic curve touches there the sheet which 
is not touched by the tangent plane a— 0 ; for this being so the quartic curve touching 
one sheet and simply meeting the other sheet meets the torse in three consecutive points, 
or the two points of intersection each count three times. 
The torse has the cuspidal line 
S=ae— 45<7-|-3c 2 =0, T=ace-\-2bcd— ad 2 — b 2 e— c 3 =0, 
and the nodal line 
6(ac— 5 2 ), 3 {ad— be), ae-\-2bd—8c 2 , Z(be—cd ), 6(ce— d 2 ) II 
a, b , c , d , e 
and the equations of the nodal line are satisfied by the values (a= 0, 5 = 0, 3 ce — 2^ 2 =0) 
of the coordinates of the points in question. To find the tangent planes at these points, 
starting from the equation S 3 — 27T 2 =0 of the torse, taking (A, B, C, D, E) as current 
coordinates, and writing 
d=Ad a +Bb 4 +C3 c -f Dd d -f-Ed e , 
then the equation of the tangent plane is in the first instance given in the form 
S 2 BS— 18TBT=0, which writing therein (a= 0, 5=0, ?)Ce — 2d 2 =9) assumes, as it 
should do, the form 0 = 0; the left-hand side is in fact found to be 9c 3 (Sce—2d 2 )A. 
Proceeding to the second derived equation, this is S 2 B 2 S + 2S(BS) 2 — 18TB 2 T— 18(dT) 2 =0, 
or substituting the values of the several terms, the equation is 
9c 4 (AE — 4BD -f 3C 2 ) 
+ 3c 2 (eA-4dB+QcC) 2 
1 8<? 3 { e( AC — B 2 ) + 2<5(BC — AD) + c(AE + 2BD — 3C 2 ) } 
- 9 {(ce-d 2 )A+2cdB-3c 2 C} 2 =0 ; 
the terms in BC, BD, C 2 vanish identically, that in B 2 is (48 — 36 = )12ck5 2 — 18c 3 <?,= 
— 6c 2 (3ctf— 2cZ 2 )B 2 , which also vanishes; hence there remain only the terms divisible by A, 
giving first the tangent plane A=0, and secondly the other tangent plane, 
A(— 6cV+18c<7 2 e— 9d*) 
+B(— Q0c 2 de-\-36cd 3 ) 
+ C( 108c 3 e-54c 2 d 2 ) 
+ D(- 36 c 3 d) 
+ E.27c 4 =0. 
Taking the equations of the quadric surfaces to be 
* (X, (*, * , g , <r, r^a 2 , 5 2 , ab, 3 ce — 2d 2 , ae—&bd, ad—12bc)=0, 
(*!,?', S,g',c rVX „ „ „ „ ) = 0, 
5? 
