TRANSACTIONS OF SECTION A. 651 
of integral configurations of the equation, 7.e. integral curve or surface is trans- 
formed into an integral curve or surface by the transformation. 
When we know, then, an infinitesimal transformation of which a given dif- 
ferential equation admits, at least one family of configurations, which (family) is 
invariant under the transformation, is of interest to us, namely, the family of 
integral configurations of the given equation. But it may happen that other in- 
tegral invariants under the transformation may satisfy the equation. That there 
are other invariant configurations is clear from the fact that a given infinitesimal 
point transformations may leave invariant a great variety of equations ; conversely 
also a given equation may admit of none, one, or several infinitesimal trans- 
formations. 
The direction (dv,, dx,,..., dx,) on the envelope of the integral configura- 
tions, since this envelope is a trajectory of the transformation, is given by the 
continued proportion 
ary a2, axn (25) 
ey aiay te 
if the infinitesimal transformation be written in the form (1); further, if this 
enyelope is to be an integral configuration of the Monge equation 
SPrareg.  eg@U AX. « AXyn=0, Ze;= mM, ° « tee te(26) 
the equation which is assumed to admit of the infinitesimal transformation (1), 
then the same system of differentials (dz,,..., dz,) must satisfy (26), that is, we 
have 
BPs REE We buen Os ol ant is QA) 
It is clear then that the equation (27) may give a singular solution of the 
equation (26), if it have one; it is also clear that no part, or only a part, of the 
locus represented by (27) need be a singular solution of (26). In case the trans- 
formation leaves every single integral configuration invariant the relation (27) is 
satisfied identically and yields nothing new. 
We have here then a method, which consists of a simple extension of Lie’s 
method for the integration of ordinary differential equations of the first order, for 
discovering singular solutions of a Monge equation without resorting to inte- 
gration. 
Furthermore, it should be remarked that not only is the method applicable to 
integrable and non-integrable Monge equations, but that nothing forbids the 
analytical application of the theorem to forms no longer homogeneous in the dif- 
ferentials, should such forms be possessed of interest or show themselves capable of 
interpretation, since it is easy to construct consistent criteria for the invariance of 
such forms under infinitesimal point transformations. 
6. Since the Monge equation of the mth degree 
SyP(tiye « y2n)esege « eg ITAL, « dann =0, Se, =m : - (28) 
is homogeneous of the mth degree in the differentials, it is equivalent to m Pfaff 
equations of the form , 
t=n 
ZW». . +0) AX; =(1)- I= Vie . aye . . e (29) 
A solution of any one of these Pfaff equations is a solution also of the Monge 
equation, and the problem of finding an infinitesimal transformation of which a 
Pfaff equation admits is obviously simpler than the resolution of the same problem 
relative to the Monge equation; in fact, there are cases in which the finding of m 
different infinitesimal transformations of which the m Pfaff equations (29) admit 
would be simpler than that of constructing one of which the Monge equation admits. 
It should be remarked that the invariance of the Monge equations under a given 
