COORDINATES OP A CUBIC CURVE IN SPACE. 
379 
It might at first sight appear that, in each of the curves (11), we should in general 
have nine points determined as follows : viz. (taking the last equation) 
by the intersection of S and S', one point, 
,, ,, T ,, S, two joints, 
,, ,, T' ,, S', two points, 
,, ,, T ,, T', four points. 
nine in all. This however is not strictly the case, for 
S', S, T', T=u, u, S, S', (u , u, suffix t; S, S', suffix 0) . . . (12) 
v, v, S, S', (v, v', „ t ; S, S', „ I) 
w, w , S, S', (iv, w', „ t; S, S', „ 2) 
and since the four determinants which can be formed from the above matrix are 
equivalent to only two independent determinants, it follows that if any two of the 
equations S = 0, S' = 0, T = 0, T' = 0, are satisfied, all will be satisfied. In other words, 
the four curves S, S', T, T', have a common point of intersection. 
Instead, however, of taking the coefficients of the equations (8) and (8'), i.e., the 
quantities of which (9) are specimens, as the forty coordinates of the curve, it will be 
convenient to take the coefficients of the functions (10) as the coordinates. These 
will be found to be forty-eight in number, and to be comprised in the following table, 
as stated in the introductory remarks. For the sake of brevity, only the upper lines 
of the determinants are written down : 
A 
A, 
a = B, 
(suffix 
*), 
/3, S', a 
+ 
S, 
/3', a = 2M (suffix 
4 
y> 
y, 
a = C, 
( 
55 
x), 
y, S', a 
+ 
s, 
y', a = 2N ( 
55 
a), 
s, 
S', 
a = D, 
( 
55 
x), 
A y', a 
+ 
y, 
A, « = 2F ( 
55 
x), 
a, 
a', 
/3=A, 
( 
55 
y)> 
a, S', ^ 
+ 
s, 
a', /3=2L ( 
55 
y)> 
r> 
y> 
0=C, 
( 
55 
y)> 
y, S 
+ 
s, 
y', /S=2N ( 
55 
y). 
s, 
S', 
£=D, 
( 
5 5 
y)> 
y, a', 0 
+ 
a, 
y', y5=2G ( 
55 
y), 
a, 
a. 
y = A, 
( 
5 5 
z )> 
a, S 
'» y 
+ 
S, 
a', y = 2L ( 
55 
*), 
A 
A, 
y = B, 
( 
5 5 
A S', y 
+ 
S, 
A, y = 2M ( 
5 5 
s, 
S', 
y = D, 
( 
5 5 
z ), 
«, A, y 
+ 
A 
II 
to 
55 
a, 
a, 
S = A, 
( 
55 
0, 
A y', S 
+ 
y, 
A, S = 2F ( 
5 5 
0. 
A 
A, 
S = B, 
( 
55 
y, a', S 
+ 
a, 
y', S=2G ( 
5 5 
0, 
y> y\ 
£XT. 
8 = C, 
( 
5 5 
a, A, S 
3 D 
+ 
A 
a', S = 2H ( 
5 5 
0, 
(13) 
