( 1249 ) 
T A 
A == anes’ 
Fg sy 2 SA a 
T Ty 
i= EN A,(A,—A,)N,N,. 
A; 
With these values and 5; = Ne the first member of (c) reduces to 
5 
op An (AoA) [204,4, — BIA HAN + 
its 
fe IN A, a) [2a4,A, =a) (A,+4,)] ie 
NE AAA AAN 
bN 
3 
= vit Ct NAA 434 aS 
Further, the first member of {e) takes the form 
or A: (AA) AA, [a (A,-+-4,) — 28] + 
T 
ae NS lar) Ar [a (A, + A,) a 2b] in 
+4, (A,—A,) A,A, [a (4,+4,) — 26] = 0 
and the first member of (/) 
mace AAA Ale 454.4, (AA) AAA (A,—A,)} = 0. 
The general integral is therefore 
(y — A,# — B). (y — Ar — B: (y — A,w — B,)s = Const. 
where 
2,:4,:4, = (A,--A,) (@A,?—2bA,—a): 
(Ae = (a= ERO — A.) @ At 26d a), 
When the cubic in A has a pair of imaginary roots the corre- 
sponding particular integrals are conjugate imaginary and therefore 
the general integral is imaginary unless two of the quantities 2 are equal. 
This is only possible if b(a + 6’) =O and these cases have already 
been considered in Art. 6 and Art. 4. We must thus suppose that 
all the roots of the cubic in A are real. 
82 
Proceedings Royal Acad. Amsterdam, Vol. XIII. 
