146 
From these six invariant functions we eliminate the differentials x’ x”, ...- 
obtaining the differential invariants: 
1 = (3 Ll, — 12, —5 1,7] : (11)3 = Bray, — Sys?) t ys! 
do= [15 Lal, 8 El, eo eS (1, — 3 1,)| :1,4 
{ 8y2?75 — loysysya +4 y athe 
15. The general linear group 
_P, 4) Xq, XP — Yq. YP, XP + yq | 
leaves invariant the quotient 
x y 1 | a y 1 
Q =| x@ y® 1]:} x@) ye) 1 
x(3) y(3) 1 | x(@) y(4) 1 
ee y i x y 1 
Q=|x+x/dt+... yty/dt+...1 >| x+x’ds+... y+y’ds+... 1 
| x+x’ds+... yt+y’ds+... 1 x+xdr-+... y+ydr+...1 
Ned fet 1 | dt ds? | + k,I, | dtds* | +-k,I, | dtds* | +k,I, | dt?ds? | +) 
~ tks I, | dt ds® | +k,I, | dt?ds*|+-k. ‘I, | adtde® | + k.I, | dt? ds® | + Ls 
| +] k,I, | at? dst | J 
; { Similar expression in ds, dr. o 
: __ | dta dtb : : 
where kj are constants, | dtadsb | — dsa dsb |» 20d Ii are functions de- 
fined asin 14. The form of this expansion for Q shows at once the invariance of 
the quotients I,:.1,, 1,:1,,....:.. . Denoting these ratios by R, we have 
R, = il Sty =(x"y LI Py x’ y’): (x’/y¥/% — x yt). 
R, =1, :1, =x’y” — xivy’): 1, 
R, a+ I, : 1 — (xy — x’ ae Oe Les 
BE, = 1:1, =«2’y' —x’y’) : 1, 
BR, = 1, 21, = (x“y"— xi"y”) : L,, 
RB, = 1,21, = (x4 — x"y’) 2h, 
R, = 1, : 1, =(x”y"—2"y”) : 1, 
R, =1,:1, = (x’y'¥ — xy”) : I. 
