406 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
251 
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 count each of them three times. 
The torse has the cuspidal line 
S — ae — 4bd + 3c 2 = 0, T — ace + 2bed — ad? — b 2 e - c 3 = 0, 
and the nodal line 
6 (ac — b 2 ), 3 (ad — be), ae + 2bd — 3c 2 , 3 (be — cd), 6 (ce — d 2 ) 
b 
d 
c 
e 
a 
and the equations of the nodal line are satisfied by the values (a = 0, 6 = 0, See — 2d 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 Avriting 
0 — Ad a 4- Bdi + Cd c 4- Ddd 4- Ed a, 
then the equation of the tangent plane is in the first instance given in the form 
S 2 dS — ISTdl 1 = 0, which writing therein (a = 0, 6 = 0, 3ce — 2d 2 = 0) assumes, as it should 
do, the form 0 = 0; the left-hand side is in fact found to be 9c 3 (3ce — 2d 2 ) A. 
Proceeding to the second derived equation, this is S 2 d 2 S+ 2S(dS) 2 — 18Td 2 T— 18(07 7 ) 2 = O. 
or substituting the values of the several terms, the equation is 
9c 4 (AE- 4BD+3C 2 ) 
+ 3c 2 ( eA — 4dB + 6cC) 2 
+ 18c 3 {e (AC - B 2 ) + 2d (BG - AD) + c (AE + 2BD - %C 2 )} 
— 9 {(ce — d 2 ) A + 2cdB — 3c 2 G) 2 = 0 ; 
the terms in BG, BD, C 2 vanish identically, that in B? is (48 — 36 =) I2c 2 d 2 — lHc 3 e, 
= - 6c 2 (3ce - 2d 2 ) B?, which also vanishes; hence there remain only the terms divisible 
by A, giving first the tangent plane .4=0, and secondly the other tangent plane, 
A (- 6c 2 e 2 + 18cc? 2 e - 9d*) 
+ B (— 60c 2 de + 36cd 3 ) 
+ G ( 108c 3 e - 54c 2 d 2 ) 
+ D(- 36c*d) 
+ E. 27c 4 
= 0. 
Taking the equations of the quadric surfaces to be 
32—2