OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 533 
§ 15. The case of § 14 presents itself quite naturally if we start from the differen- 
tial equations (I.) instead of from the integrals (II.). Let ¢, as before, be a factor 
of E. 
There is no apparent reason why the equation 
should follow algebraically from the system (I.) and the equation ¢ = 0. 
Let E, = 0 be the result of eliminating p,, p,... p, from the system (I.) and the 
equation d¢/dax = 0. 
Let ¢, be a factor of EK, and let E, = 0 be the result of eliminating p,,... p, from 
the system (I.) and d¢,/dx = 0, and so on for ¢3, ¢,... 
There is no apparent reason why any function in the series ¢, ¢, ¢,... should 
vanish because all those before it are supposed to vanish. If one of them, say ¢,, does 
satisfy this condition, then the equations 
are integrals of (I.), and by using them and finding  — r other integrals, each con- 
taining an arbitrary constant, we have a singular solution of the r” system. 
§ 16. Thus, as in the simpler case when there is a single equation of the first order, 
the existence of singular solutions appears to be the rule if we consider the integrals, 
the exception if we consider the differential equations, and in fact the number of 
conditions for the absence of the first r singular solutions rises with r from the first 
point of view, while the generality of the conditions for the existence of the (7 + 1)™ 
from the second point of view decreases as 7 increases. 
of 4, C2... C,, but that when 2, y,...y, are eliminated from (VI.) by this means, an integral equation 
presents itself as an alternative to a differential equation. In such a case the integral equation will, of 
course, relate to the second singular solution, for it contains no arbitrary constant. 
Considered geometrically, the second singular solution thus arising will be enveloped by the complete 
primitive, and, therefore, also by the first singular solution (for both he on the locus of singular solutions 
E = 0), although it may not be a singular solution of the differential equations (VI.). 
A proper change in the forms of the arbitrary constants would reduce this case to the ordinary one. 
If the integral factor is 61 ¢*!..., 0,6... being functions of ¢,c,... thenif in the system of 
arbitrary constants we take 6” ¢’.. . instead of c,, the factor will disappear. 
In fact the ordinary equation 
y = pz — p*, 
or any other, may be transformed so that its singular solution appears as an integral factor. Put 
ey 7a 2 
y = 1 (@ — 2) 
and we find 
