532 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
be different, since the singular solutions of the system do not furnish the same 
values as the complete primitive for any but the first differential coefficients 
(compare § 17 below). 
Singular Solutions Included in the Complete Primitive. (SS 14-16.) 
§ 14. In certain cases it appears to be possible for the second singular solution to 
exist without the first. 
The ratios de, : de.... are given by the equations (VI.) and involve a, y;, Y. . - Yn 
which are given in terms of ¢,, c,... by the equations (II.) and (VIL). If these are 
not enough, that is to say, if © can be expressed in terms of ¢), c,... ¢,, there is no 
first singular solution. 
For, from © = 0, may be deduced 
n ALO) 
> ee dco —10) 
7=1 OC, 
a further linear equation to be satisfied by de,, dce,,... Thus either ¢), c,... are all 
constants, or the further integral equation 

whose left-hand side we shall call 9), is satistied. 
In virtue of this equation © = 0 is an integral of the equations (VI.). The values 
of x, y,... Y, are given in terms of ¢,,...¢, by (II.) and 0, = 0, and by substituting 
these values in (VI.) and finding 7 — 2 more integrals of the equations so derived, 
we get the second singular solution, containing 7 — 2 arbitrary constants. 
It would, perhaps, be better to say that in such a case the first singular solution is 
included in the complete primitive, the values of the arbitrary constants being so 
chosen as to satisfy the equation 2 = 0. 
If Q, can be expressed as a function of c,,...¢, only by means of the equations (II.), 
we must use the equation 
ONE BO) 

to find the third singular solution, the second being included in the first. 
Because 2, = 0, 2, = 0 is an integral of (VI.), and because 0, = 0, 9 = 0 is an 
integral. Therefore 7 — 3 integrals are still to be found. 
This process may be carried on as far as is needed. It is also to be used if 0 has 
any factor that does not involve a, y, ... Yn.” 
* It sometimes happens that the equations (I.) and (VIL) are enough to define x, y;,... y, in terms 
