COORDINATES OF A CUBIC CURVE IN SPACE. 377 
The system (4) will then take the form 
Vjf\~ St -j- 6u t -\-9iS = 0 , 
v i -\-S l t-\-9vf -\-9tS 1 '=0, 
Wf S. 2 t -J- Ow/ -j- 9tS 2 == 0 ; 
or, as it may be written, 
0 = u -\-St-\-9u -\-9tS' (u, u , suffix t ; S, S', suffix 0) . (6) 
0 = v -\-St-\-9v' -\-9tS' (y, v, ,, t; S, S', ,, 1), 
0 =w-\-St-\-9w' -\-9tS' (w, vf, ,, t; S, S', ,, 2), 
and if we multiply these equations throughout by t, we may write the two systems 
(6), and t(6) thus : 
u-\-9u' -\-tS -\-9tS' . . . =0, (u, u. 
suffix t; 
S, S', suffix 
0) 
v J r 9v'-\-tS-\-9tS' . . . =0, ( v , if, 
3 5 ^3 
S, S', „ 
1) 
w-\-9w'-\-tS -\-9tS' . . . =o , (w, w , 
„ t; 
S, S', 
2) 
. tu-\-9tv! if S-\-9t?S'— 0 , (u, u, 
33 ^ 3 
S, S', „ 
0) 
. tv -\-9tv +^S+^ 3 S'=0, (y, v, 
„ t; 
S, S', ,, 
1) 
. tw-^-9tw'-\-t 2 S-\-9t z S'=0, (w, vf, 
33 i 3 
S, S', „ 
2) 
whence we may at once eliminate 1, 9, t, 9t, t 3 , 9tf, and obtain the following result: 
u, 
u , S, S', . . 
= 0, ( u , u, suffix t ; 
S, S 
', suffix 
0) 
V, 
v, S, S', . . 
(v, if, „ t; 
S, S 
? 5? 
1) 
w, 
vf, S, S', . . 
(vj, vf, „ t; 
S, S 
5 
2) 
. u, u, S, S' 
(u, vf, „ t; 
S, S 
/ 
3 33 
0) 
. v, v, S, S' 
(v, v, ., t; 
S, S 
/ 
3 33 
1) 
. w, w', S, S' 
(w, vf, „ t; 
S, S 
„ 
2) 
The corresponding results, when x, y, z are respectively eliminated, are obvious; viz.: 
writing down the upper lines only, they may be represented thus : 
U, U , a, «', 
! =0, (u, u, suffix x) 
u, u, fi, (3', . 
3- 
o 
II 
U, U, y, y , . 
| =0, (u, vf, ,, z) 
(S') 
From these equations it is not difficult to write down the coefficients of the powers 
and products of the variables in each case. Thus in the case of (8), or t eliminated, 
