( 1248 ) 
a, +4, HA mie 
AA, HAA, + 4,4, = — Or 
1,4;4, + MAA, Hr a,4,A, = ar. 
Hence 
— 5 (A,—A,) (a4,?—2.A,—a) 
% 2 
4, = yr Ai) (24,*—284,—a) 
T rn 
A == ay (Ai A) (aA,’—2bA,—a) 
where 
N= AA AS AN Ay Ae HA 
and it is still to be proved that the three conditions (c) (e) and (/, 
are satisfied by these values of 2, 2, 2,. Before proving this, we will 
write 2, 2, 2, in another form. 
The values of B, B, B, expressed in the values A, A, A, are 
found to be 
B, = o(ad,* — 2bA, — a) 
B, = a(aA,? — 26A, — a) 
B, = o(aA,? — 2bA, — a) 
and it is not difficult to find the value of 6. For introducing 
cj bB 
Vi GB 4 
in the cubic, it is evident that the values of B are the roots of the 
cubic 
2a(ab! — b*) B® + (3b? — Aab' — a°)B' + Ua + 6) B—1=0 
Therefore 
3b°>—4ab'—a’ 
2atab' —b’) 
and 
1 
= 560d) 
Now 
aA, — b)(a A, -- b) (aA, — 6) = N,N,N, = 
— aA AA, —a°b(A, A, + AA, + A,A,) + ab(A, + A, + A)—8? = 
b 
—= 206(b?—ad') = - 
and finally 
