140 
The series expansion of v, is identical with that of v, in 4 above. Hence, 
the invariant functions may be written 
x’ x// xiv I en y’ AS Yi ,% // 4/ 
x’? ee ee 1 (x’)e (x’je-1 ext a (xe (x’)er1 
Van 5 kG 
(@wyee TB G7 
SSS Sed // v // 2 S// 
I,= 2 7 a 7° es Ts = pe Bt 
Oe er oe (CxO) x Ee rx 6 x 
x” x rx) 2) yi 
yen thar Ge (bo et ke er) fee 
From these relations follows at once the invariance of 
Bde Y2 Y3 
(x Kea? (x)= 2 (x )e— 8: 
By eliminating x’, we have 
y— y”) 
—y" fe 
6. | q, yq | leaves invariant x and vy, = Expanding v, in series, 
f dr? ees dt? ) 
Vg =34yY—(y+y’ dr+ vy” —— Jrisy—(y+y’ dt ees ey 
dr gir: yr” Or i AGP ee 
— — ———dr? | — + 
dt 2. 9 Qy’ J : 
y” se 
which gives invariant functions —,—, ..... The functions x, x’, x” .... 
7 ay, 
are also invariant. 
we) y: 7 ig 
h == 5 
ee tga ee 
Y2 
y= x eo — a 
7. The group 
q, ¥9; P 
has point-invariants 
y — y(3) 
u, = x— x(2), v, => _ 
. pap 
