384 
MR. W. SPOTTISWOODE ON THE FORTY-EIGHT 
From this we may conclude the following group, 
2d?= I a, (3, 8 I (I), z, x)+ I y, a, 8 I (D, x, y) . (18) 
2<£»= | a, /3, 8 | (D, y, z)+ | f3, y, 8 | (D, x, y), 
2$= I r> a > S I P, y, z)+ | A y, S | (D, z, x). 
From (17) and (18) we can now eliminate the three quantities \/3, y, S|, |y, a, S', 
j a, (3, 81, and obtain the following identical relations, 
B(D, x, y)~ — 2^(D, x, y)(D, z, £c) + CT(D, z, x) 2 =0 . . . . (19) 
®(D, V, 2 )'- 2 ©(D, y, z)(D, x, y)+&(D, x, y)~ = 0 
&(D, z, xf— 2$|(D, z, x)(D, y, z) + B(D, y, z) 2 = 0 
to these may be added 
— Jp(D, y, z) 2 —St(D, z, x) (D, x, y) + $(D, x, y) (D, y, z) +©(D, y, z) (D, z,x )=0 . (20) 
— (Et(E), 2 , x) z -\-ffi(D, z, x)(D, x, y) — B(D, x, y){ D, y, z)+dP(D, y, z)(D, z, x) = 0 
—3&(D, sc, #+®(D, z, F)(D, a-, y)+&(T>, x, y)(D, y, z) — (B( D, y, z)(D, z, x) = 0 
But, as the six equations (19) and (20) are in any case equivalent to only three 
independent conditions, it is not necessary to go beyond the equations (19). 
[Postscript. 
Added April 23, 1881. 
In the present case however the equations (19) are themselves not independent, as 
may be shown in the following way. Write for the moment 
D-X, D,=Y, D : =Z ; D/=X', D/=Y', D/=Z'; 
then 
(D, y, z) = Y'Z-YZ', (D, z, x) = Z'X-ZX', (D, x, y) = X' Y-XY'. 
Inserting these values in the first equations of (19) we obtain 
B(X'Y - XY'Y - 2&(X'Y - XY') (Z'X - ZX') + £T(Z'X - ZX') 2 =0 
= (BY 2 +2ifYZ+ @Z~)X'~ + BX 2 . Y' 2 +<PX 2 .Z' 2 +2# X 2 . Y'Z' 
- 2 <&ZX. Z'X' —2 BXY. XT'— 2dfXY. Z'X' - 2tfZX. XT'. 
But 
BY 2 +2^fY Z + &Z 2 
= (CX 2 - 2GZX+AZ 2 ) Y 2 - 2 (A YZ - HZX - GX Y+FX 2 ) YZ+(AY 2 - 2HXY+BX 2 )Z 2 
= (CY 2 —2F YZ+BZ 2 )X 2 
