528 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
We have also identically 
2 
Gre) i=0 C2 Woop 1, 
\ 
and therefore 
Thus a 
(= ap } (AyA + Aah =P 50557 UN) = 0, 
where A,, A,... A, are arbitrary functions. 
These may be so chosen that SAf is the eliminant of f,, f2...f, with respect to 
P2,-++P» We will call this eliminant P,; it does not involve p,... p,. 
The equations then become 
y 
(ng) 2 =° (Sie ee) 
\ / 
that is to say 
ie) pes 0 (Gai b. . Ph 
Thus the equation P,; = 0 gives 7+ 1 coincident values of p,. The same holds 
OIE (Bey, Vx 3 oo JD 
If we take a system of values of a, y;,...y, such that s consecutive members * 
of the 7" singular system of solutions are satisfied, then s — 1 consecutive members 
of the (7 + 1)™ will be satistied, s — 2 of the (7 + 2)" and so on to the (7 + s — 1)", 
after which none are satisfied. Also s+ 1 of the (7 — 1)" system will be satisfied, 
s+ 2 of the (7 — 2)" and so on, and lastly s + 7 of the complete primitive system. 
§ 7. The second singular solution of (I.) is the first of (VI.) and (VII.) Now 
by (VI.) the ratios de,:dce,:...:dc, are given rationally in terms of ¢,, cy... Cy, 
and certain other variables x, y,...y, which are connected with c,...c¢, by the 
equations (II.) and (VII.) The condition that (II.) and (VII) shall have two 
consecutive solutions is 
© being written for the left-hand side of (VII). 
This equation we may write 
) é ho Wai an 
(a, + Pig, +225, + ne | Oj —0ROG eNO =—s05 
oY 
* Different members of the system are got by giving different sets of values to the arbitrary 
constants. 
