380 
MR. W. SPOTTISWOODE ON THE FORTY-EIGHT 
with a similar set of expressions of which the following are specimens 
/3, (3\ a' = Bh, (3, S', a! + S, /3', a' = 2W J . 
. (14) 
Substituting these values in (10), we shall find the following to be the forms of the 
functions therein contained, viz. : 
U =(B, C, D, N, M, F )(y, z, tf, 
V =(C, A, D, L, 1ST, G)(z, x, tf, 
W=(A, B, D, M, L, H)(x, y, tf, 
T =(A, B, C, F, G, H)(x, y, zf, 
B =A I/ y J r A.z +A ( t, 
Q = B^x+B ; 2 + B^, 
II =C_ l x-\-C 1/ y fi- CV, 
S = D u x fi- ~D y y + D.z, 
U' =(B', . . . ) (y, z, tf; (B, . . . suffix x) 
A' =(C', . . . ) (z, x, tf; (C, . . . „ y) 
W / =(A / , ...)(x,y, tf; (A, . . . „ z) 
T =(A',. . . ) {x,y,zf; (A, . . . „ t) 
B =A/y+ ■ ■ ■ ) 
Q — B Jx -(-...) 
FF = C,'x + ...) 
S = D. r x+ . . . ) 
(15) 
And finally we may, by means of the equations (11), replace forms having the 
coefficients (9), by forms having coefficients into which the quantities A, B, . . . enter. 
Thus, writing 
B j A;-B.,'A,= (B, A, x, y), CU,'— C/A,=(C, A, x, *),... 
The forms in question are 
{(B, A, x, y), .(15') 
(C, A, x, z), 
(D, A, x, t), 
(C, A, x, £) + 2(N, A, x, z), 
(D, A, x, z) + 2(N, A, x, t), 
(B, A, x, 0 +2(M, A, x, y), 
(D, A, x, y) + 2(M, A, x, t), 
(B, A, x, z) + 2(F, A, x, y), 
(C, A, x, y) + 2(F, A, x, z), 
(N, A, x, y)+ (M, A, x, z)+(F, A, x, t)}\y, z, tf 
