652 REPORT—1899. 
transformation by no means carries with it the invariance of a factor equation, nor 
reciprocally. The invariance is reciprocal only in the case where the Monge equa- 
tion can be broken up into m equal linear factors. 
These observations bring into light the double usefulness of the method as 
applied to finding singular solutions of Pfaff equations. 
In the latter particular form it is, again, applicable both to integrable Pfaff 
equations and to non-integrable ones, and thus generalises! a theorem of Guld- 
berg’s inserted in the ‘Oomptes Rendus’ of December 26, 1898, to the effect that 
linear integrable total differential equations can admit of singular solutions whose 
determination can be effected without integration. Guldberg’s theorem makes no 
reference to Lie’s theories in its statement or demonstration. The corresponding 
theorem for ordinary differential equations of the first order was given by Page.” 
In Guldberg’s later memoir already referred to the classic theory relative to 
singular solutions of ordinary equations is extended directly to total equations of 
the first order, and first and second degrees, without use of Lie’s methods. 
7. Before appending a few concrete examples we shall find a new interpretation 
of the (_) operation of Lagrange and Poisson which plays so capital a rdéle in the 
theory of continuous groups; it came to light in constructing a new proof of 
Guldberg’s theorem above referred to. 
Consider again the non-integrable linear total differential equation in three 
variables. 
P(ayy,z)dx + Q(a,y,2)dy + R(a,y,2)dz=0. : : . (80) 
An integral of this equation satisfies the linear partial differential equations 
uf= F_ PY _o, 
ox Rodz 
(31) 
—dy Roz 
in order that the system (31) have a solution it is sufficient that the relation 
(U,V) = UVf — VUf =0 : 5 ; . (82) 
wea of WW _o, 
shall be satisfied. 
Developing this relation (32) we find 
(U,V) = - et P(Q:—R,) + Q(Re— P:) + R(Py — an of = 0;.. -(88) 
the hypotheses 
R =co,f, =0 
exclude themselves, then we have 
PQ, —R,) + QR. —P,)+R(P-Q)=-0 . . 
The solutions s = f(2,y) of this algebraic equation which satisfy the non- 
integrable equation (30) are singular integrals of the latter. 
If the equation (30) is integrable the relation (34) becomes an identity, and (33) 
furnishes the interpretation of the ‘ Klammerausdruck’ of the infinitesimal trans- 
formations Uf and V/ above, namely, that its vanishing expresses the condition, 
necessary and sufficient, that the linear total differential equation (30) shall be 
integrable. 
8. Examples. 
1 Comptes Rendus, 31 juillet 1899. 
2 American Journal of Mathematics, 1896. 
