DIFFERENTIAL INVARIANTS OF SPACE. 
295 
Thus il is unaltered when n and v are interchanged ; it is changed to Q, when 
V and IV are interchanged, and to P, when u and w are interchanged. The 
interchanges of a, b, c among one another, and likewise those of f, g, h among one 
another, caused by interchanges of the variables u, v, w, are indicated in the table ; 
they could be used to deduce b and c when a is known, and to deduce g and h when 
f is known. 
1(3. We have now to obtain live functional combinations (other than Id and 0) of 
the fifteen quantities a, h, c, f g, h, ^oio’ ^oou S'’ ''vhich satisfy the 
nine equations 
Ai (cr) = 0, . . . , A(; (o-) = 0, 
A;(c 7 ) - A8(cr) = 0, A~ (cr) - Ag (cr) = 0, A-(cr) + Ag (u) + Ay (cr) 3/xcr ; 
and the functional combinations which are required must contain some of the 
quantities a, b, c, f, g, b. 
It is easy to verify the results m tlie following table : 
a 
b 
c 
f 
g 
b 
Ai 
2 I 1 
0 
u 
0 
f 
b 
Aj 
2g 
0 
0 
0 
c 
f 
A;i 0 
2li 
0 
g 
0 
a 
A4 
0 
2f 
u 
c 
0 
S 
A5 
0 
0 
-'g 
h 
a 
0 
A.i 
0 
U 
2f 
b 
0 
A; 
la 
2b 
2c 
2f 
•3g 
3h 
As 
2a 
lb 
2 c 
;if 
2g 
3h 
A, 
2a 
2b 
Ic 
•3g 
2h 
which should be read A^^ (a) = 2li, A^ (b) = 0, and so on. Pleuce now denoting by cr 
any one of the five functional combinations ol the fifteen arguments, the first eight of 
the equations take the form 
