650 REPORT—1899. 
_4. Seeking now the variation of the Monge equation (9) due to the infinitesimal 
point transformation in three variables 
Of , of , OF 
é kr ane, . . . . . (18) 
as was done for the Pfaffian (6) in the preceding paragraph, we find 
5M, = Pddz* + dx*8P+ ... +Sd(dydz)+dydzdS+..., 
= 2Pdrdéx + dx*(P,ba + Pydy + P.dz) +... + S(dyddz + diddy) + dydz(Sz2x + 
S,dy+8.82) ....3 
observing as before that (18) gives to 2, y, = the respective increments 
dz = £58, dy = dt, dz = (dt, 
substituting and neglecting the factor 8¢, we have, after an easy reduction, 
6M, =a? + dy? + pdz? + 2Qadydz + 2rdzdx + Qudxdy, . - (19) 
where 
n= 2(P£,+ Un. + TC) + EP, + nP, + CP., 
K=2(UE, + Qn, + SG) + EQ. + nQy+ (Quy . 
p=2(TE, + Sy. + RE) + ER, +R, + CR., 
o =TE, + UE, + Syy + Qn. + RG, +86 + £8, +78, + 6S. 
Tr PE, as TE, at Un. at Snz = NG + RG + ETT, + Gi 
v=UE, + P&, + Qnz + Uny + SG + Tg, + £U,+qU, + (U.; 
(20) 
hence the Monge equation (9) admits of the transformation (18) if the following 
conditions 
hold. 
By the same method the invariance criteria for a Monge equation of the second 
degree in x variables may be found. If the equation is 
$=2M,,(2,,..., tn )dridx =0, t,j7=1,...,2, M,;=M,,> - (22) 
the variation of ¢, due to the infinitesimal point transformation (1), is found 
to be 
mj; dxjdxj, . 3 : s ; » (28) 
where 
t=n, j=n, k=n 
= (Mage, + Mixés, ). . » (24) 
j=1, k=1 i oj 
h=n 
Mi — = &.Mi,; ar 
h=1 i=1,j 
Then the necessary and sufficient conditions that the equation (22) shall admit 
of the infinitesimal point transformation (1) are expressed by the equality of the 
quantities 
Mi 5 
M. 
uJ 
where both 7 and j take all values from 1 to m, and both may take the same value. 
It may be remarked, in passing, that these forms show that not every Monge 
equation admits of an infinitesimal point transformation; they indicate at the same 
time how complicated the invariance criteria become for equations of higher degrees 
and higher orders. 
5. The geometrical expression of the invariance of a differential equation under 
an infinitesimal point transformation is that the latter leaves invariant the family 
a 
50h ase 
—— 
Pe ainty 
