OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 539 
§ 24. In order to find the other second singular solution let us write T, U, V for 
(wu — v) (1 + 2) (@-+ tA) (6 + ¢B) (c + tC), and the two symmetrical expressions. 
The differential equations, cleared of fractions, may then be written 
du 
UVuv + VTvt ey 4 TUt (Fp = 0, 
dt 
UV + vase hs mul) = 0. 
The Jacobian of these expressions with respect to du/dt and dv/dt is 
4 du/dt dv/dt T?UVt (vw — w), 
the square of which reduces to a constant multiple of 
T?UPVeuv (¢ — u) (t — v). 
The factors of this expression are to be considered in turn. 
Now the vanishing of such a factor as 1 + ¢ or a + At only causes two solutions of 
the equations giving a, ¥;, yz in terms of t, u, v to coincide and only yields a solution 
of the transformed, not of the original, equations. 
The solutions given by supposing ¢, u or v to vanish or be infinite have been con- 
sidered. The case when two of the three are equal is left. 
If ¢ = u, then V = 0, so that the equations give 
TU (dvjdt)? = 0. 
Suppose first that v is constant. 
It is easily found, as in the theory of confocal conicoids, that 
Paces (6 — B)(«—C) (a + At) (a + Au) (a + Av) 
~ Aa(aB— Ab) (aC — Ac) (1 +4144) (142) 

x 
with like values for y,” and y,”. 
Thus if ¢ = ~ and v is constant, x, y,, yz are constant multiples of 
m@ sc Me oe BE @ St Ce 
jeer en We eg 

Hence y, and y, are linear functions of « and p,, py are constants connected by a 
single relation since they involve the arbitrary constant v. This solution is therefore 
included in the complete primitive and must be the same as was rejected in § 22. 
3 Z 2 
