COORDINATES OF A CUBIC CURVE IN SPACE. 
385 
also 
£TZ+d?Y 
= (AY 2 —2HXY+BX 2 )Z —(AYZ —HZX—GXY+FX 2 )Y 
= (BZX - FX Y - HYZ+GY 3 )X 
= -<0X 
and 
BY+i?Z 
= (CX 2 - 2GZX+AZ 2 ) Y - (A YZ - FIZX - GX Y+FX 2 ) Z 
= (CX Y - GYZ - FZX+ HZ 2 ) X 
=-m 
Hence the whole expression 
= (&X' 2 +BY' 2 +CTZ' 3 +2^Y'Z'+2(GZ'X / +2$?X'Y')X 2 , * 
or 
(«, as,.. . )(X', Y', z') 2 =o 
i.e., 
A, H, G, D,D;=0.(21) 
H, B, F, D y , D/ 
G, F, C, D„ D/ 
U f , D y , D,, . . 
d/ 3 d/, d/, . . 
The equations (19), and consequently also the equations (19) and (20), are therefore 
together equivalent only to the single condition (21). 
Now the relation (21) has been derived from the form T; if to this we add the 
corresponding relations derived from the forms U, V, W, we shall have four relations. 
Others are readily obtained as follows. If we form the four systems of which (17) is 
one, we may write them down thus :— 
33 = | 7 , a, S 
(A, z, t), ©= I /3, 8 1 (A, y, t), 13 = 
I «, ft 71 (A, y, z), (suffix x) 
& = I ft, 7. £ 1 (B, 
t), 
(§ — \u, /3,8 \ (B, z, t), 13 = 
ft 7 1 (ft 2, ( 
» y) 
7,S|(C, 
y, t), 33 = 1 7 , a, S | (C, x, t), . D = 
I «, ft 7 1 (C, a, y), ( 
>> 0 
a=|A%8](D, 
y, *), 33 = ] 7 .«, SI (D , x )> €=\u,/3,S\ (D,x, y), 
( 
t) 
Whence 
& 
: (B, 2 , t)=% : (C, y, <)=« 
: (I), y, z ), . . 
(22) 
B 
: (A, z, t) = 
= li : (C, a, 0 = 
: (D, 2 , x), 
<E 
: (A, y, t) = ® 
: (B, x, t)= . = (F 
: (D, x, y), 
U 
: (A, y,z)=JB 
: (B, z, as) = 33 : (C, cr, y) = 
■ 
(suffix x) 
(suffix y) (suffix z) 
(suffix t) 
We thus have eight more relations. And if to these we add the corresponding 
relations derived from the forms IT, V', W', T', we shall have 2(4 + 8) = 24 relations in 
addition to (16), viz., 8 + 24 = 32 in all.] 
