526 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
If not, that is if the two sets of values of p,,... are different, all the determinants 
of the matrix 



Ox; SE Ou eas tan Oy ii 
one om | 
Ox Oy) | 
OF, oF, | 
Be 5.0.0 On | 
must vanish (compare § 11, below.) 
The equations (III.) are then not enough to determine p,, p,...p,, and the 
conditions (VI.) do not ensure that the system of equations found by the above 
process will satisfy (I.) and be a solution at all. 
In general, then, for a singular solution, the two sets of values of p,, py... p, are 
coincident. But these are given by the equations (1.). 
Hence generally the equation 
2 Op ot ee (Valles 
O (Py; Pos - + » Pn) 
is satisfied by a singular solution. 
We have, therefore, the following process for finding the singular solution from the 
differential equations (I.) :— 
Form the equation (VIII.) and let E = 0 be the result of eliminating p,, po, ... Da 
from (I.) and (VIII.). Suppose that ¢ is a factor of E such that the equation 
d¢/dx = 0 can be deduced, by substitution without differentiating, from ¢@ = 0 and 
(1.). Then, by treating ¢ = 0 as if it were a particular first integralt of (I.), which is 
now allowable, reduce (I.) to a system of 7 — 1 differential equations in 1 variables. 
The complete integral of this system belongs to the first singular solution of (I). If 
is the only factor of E that satisfies the condition of being an integral, then it yields 
the whole of the first singular solution. 
The first singular solution of the new system of 7 — 1 equations gives the second 
singular solution of the original system of m equations and so on (see also § 15, 
below). 
* Mayer finds this equation as the condition that the second and following differential coefficients, as 
given by the equations (I.), should be indeterminate, and thus shows that (VIII.) must always be 
satisfied by a singular solution. 
+ If the phrase “first integral” is restricted to such as involve only one arbitrary constant, we may 
use “singular first integral” for such an equation as ¢ = 0 is here supposed to be. 
