TRANSACTIONS OF SECTION A. 649 
An integrable Monge equation may have three kinds of integrals, just as an 
ordinary differential equation of the first order, for example ; and the terms general 
integral, particular integral, singular integral have the same signification when 
applied to integrable Monge equations as when used with reference to an ordinary 
ditferential equation; a singular integral is one which is neither general nor 
particular. 
A non-integrable Monge equation may have singular solutions; by the latter 
we mean relations in the variables 2,,...., %», which satisfy the equation. 
3. The criteria for the invariance of the Pfaffian equation (5) under the infini- 
tesimal point transformation may be found by a simple reckoning. Thus confining 
attention for convenience to the equation (6) in three variables we have 
Om = dxbP + Pdda + dy8Q + Qddy + d2bR + Rédz, 
= dxdP + Pddx + dyiQ + Qddy + dzdR + Rddz, 
by the commutative property of the operations d and 6. Substituting the values 
(2) of d2, dy, dz assigned by the given infinitesimal transformation (1), and neglect- 
ing the factor d¢, we have 
om = dxbP + Pdé + dy8Q + Qdy + dzdR + Rd 
= da(P, 6x + P,dy + P.dz) + P(E,du + Edy + E.dz)+... 
=du(P2§ + Py t+ P:) + P(Ecda + Edy + E.dz)+....  « (10) 
Then the invariance demands that 
or =0 F ; ; : . (11) 
as a consequence of 
7=0; 
hence the criteria are 
Ly Ras Seed . 2 
P-O°R’ ; : : 3 . (12) 
where 
w=P,€+ Py + P.f+ PE, + Qn.+ RE,» 
Kk=Qzr€ + Quy + Q2€4+ PE, + Qn + RG, U 
p=Ri€ + Ryn + R.6+ PE. + Qn. + RE. 
Similarly for the equation (5) in z variables to admit of the infinitesimal point 
transformation (1) we find it necessary and suflicient that the quantities 
(18) 
‘S$ (-0P. pOki\p aL 
x big,) + Pig )IRs fei a. songeneat char Re (14) 
shall all be equal. 
__Sometimes the calculations are facilitated by making use of the system of partial 
differential equations : 
Hie BL bn py AE hyn hes oxrmcsynt 685 <x! 1B} 
equivalent to equation (6), and of Sophus Lie’s criterium for the invariance of a 
system of linear partial differential equations (complete or incomplete) 
5" eee 
ViFSX Xp i(ey sey In) = 0, JH=1, 2,4. 47, : - (16) 
under the infinitesimal point transformation (1), namely 
(U, V )=2p (2... an) Vif, (17) 
where 
(U, V,=UVif- V;Uf. 
