120 
respectively, where ot is an infinitesimal independent of x and y. Such infinites- 
imal transformations move a point x, y through a distance 
Vix? $ by? = Fa at, 
and in a direction given by 
dy Ox ae: 
The variation of any function ¢ (x, y) by this infinitesimal transformation is 
given by dé dé do a9} * 
66——? bx 4 © by — be dO) 5 3 
- The variation of a function f (x, y) may be taken as a definition of an infin- 
itesimal transformation; in the Lie notation we have an infinitesimal transforma- 
tion defined by 
eek di di 
AE SSF py) Ge ty) = 
If a function ¢(x, y) is to remain invariant by the operation Xf, then the 
variation : b¢ (x, y)=X¢.dt 
must vanish. Hence, a function % (x, y) invariant by the infinitesimal trans- 
formation Xf is determined as a solution of the linear partial differential 
aa Xf=f Gy) Tin (sy) 20. 
The infinitesimal transformation X f may be extended to include the increments 
of the co-ordinates of any number of points xi, yi, (i—=1,2....n). We shall 
write this extended transformation thus: 
7 a | 7 (i) me . | end dt df 
W f = SiX Of = diff (xs yd) ae, +7 (x6 ¥1) = 2, ef eee 
1 1 i Yi 
The functions of the co-ordinates of n points invariant by Wf will be the 
2 n—1 independent solutions of Wfi=0. n of these solutions may be selected 
in the form (xi, yi), where ¢ (x, y) is a solution of X f—0; the remaining n-1 
solutions will in general differ from ¢ (x, y) in form.T 
r infinitesimal transformations X,f, X,f .... Xrf are called independent when 
no relation of the form 
c, &, fe, KAI sa kee + erXrf=0, (ci=const.), 
exists. If r independent infinitesimal transformations Xxf, (k—1...r), be so 
related as to form a group, then will 
Tr 
he (X,f)— X, (X,f)===s ciks Xf, (Ciks—constants) . . . (3) 
1 
“Throughout this paper df df are employed to denote partial differentials of f with re- 
dx’ dy 
Spect to x and y. 
+ Lie: Theorie der Transformationsgruppen, Bd. I., 259. 
