138 
which show x9 507/p.0.50. py oY 52 i«e, to. be invariant: . But Y=yix, y”= 
y2(x’)?-+y,x’”’; hence, y,, y. must each be invariant. 
ete ——F ty Pa Yai 
3. The point-invariants of the group 
\ a xpt+yq | 
x (2) — y®) 
are ge are Boy 2 
x P 4 
Introducing the series expansion of x(2), y(2), 
uy — (x x/dt-} x” Bo) foxss 
dt? 
= {y—(ytyat+y” > + <a 1x. 
u, shows the ratios 
4/ S// 
Kn oe 
er ge, ie ere nop kin ke =a Ieee ee (1) 
to be invariant, while v, requires the invariance of 
7 4/ =" hbk 
anes. 
x , 
Y lh 
x’ > wile 
oo ecg eda 
| aed ao ’ 
x x 
hence y, is invariant on account of (1). 
ty Za x’ / 2 
y fey +y,x x 
== = Ya His a els poral. — 0, = XxX} eg F 
>< D4 
x x 
Therefore, ¢, = y;, $, = xyz. 
4, The group 
Pq, xp+yq | 
has the point-invariants 
x — x) x—x@ 
One differential invariant may be computed from u, alone, but a second cam 
not be had on account of impossibility of the elimination of the parameters. We 
therefore consider three points determined by t, r. 
] dt? 
ws =jy-(ytyat+y”4 ie —4+. minal ai 
a y’ dt yy” y’ x” dt2 y’ (x/’)2 y’ x’ 
lor 4 T DN aro ne (rhe )+ 2 CAGES oe rat BD) 
