( 1247 ) 
powers of zv in both members, we have 
A (2bA — aA? + a) = WA — DA? + b 
A(26B + A — 2aAB) = 26'B — 2bAB — 1 
AB (1 — aB) = — 6B? 
which are satisfied by the roots of the cubic 
gA* — Shs IDD! — a) bd, 
and by 
EE! 
aA—b 
In this case, the general integral may be written 
(y Ge yy) (y FI y‚)” (y Ei 1)’ = const. 
where ¥, 7, y, stand for the three particular integrals and 4, 2, a, 
are certain constants. To prove this we will show that the necessary 
and sufficient condition '), that 
À À 
1 2 3 
ent mm Is 
is divisible by y + aa? + 2bay — ay’, may be fulfilled by choosing 
properly the constants 2, a, 2, ° 
iY) (yy —= 9) (ah =~ Ys) ; 
This condition may be written, t being a constant factor 
A, +2, de — av 
A(Ys + Ys) + Ao (WH Ys) + Aly, Hy) = —(L + 2ba)r 
AYaYs + AsY1Ys + Asis = ART 
or, replacing y, 4, y; by their values 
(a) A, HA, HA, = — at 
(6) AA, + A,) + AA, HA) + (A, + A) = — Zr 
(AB: + Bi) + (B, Bi) +a HB) te 
(AAA 2.4.4, AAA ar 
OA Be EAB) BAR. RAB) JMA Bid AB 0 
(7. 228, ADR 4. 1B Be 
As 
3b 2b. a b 
BNA AA MA Al AA 
a 
a a 
the conditions (a) (6) and (d) may be written 
1) KoRrKtine, Math Ann. 48, p. 350, 
