141 
We have, as in 6, the invariant functions 
Za / yy 
x6 Vox >.< 
vA f/f SSS 2 J 2 
Xe ex: soon Ly = 7 = a ) 
b] ? > ’ 1 y’ Yi x’ 
yy Y; (x’)2 : Jax’ e da 
I, = = =F : 7 I 7 
M/ Yi yi x 
Vv 
ee ae Y3 
ft Yai 
8. The point-invariants of the four-parameter group 
i eS yq 
x — x) y y?) 
eae a3 x — x’)? oy y— yo 
The series expansion for u,;, v, in powers of dt, dr will be identical with 
those for v, in 4 and 7, respectively. Hence, we have the invariant differential 
functions 
: > A Map 24 
QI) GT) GP Crete t eee eens Pai) | 
and 
tf jae Sf 1f / / ff 
et eR Ee aR 9, gYaw A ee 
SS 2 SS a 1 oe" r 4) 
iy y1 x yi Yi yy me 
I Eye Vax SG Oval Ee)? en Y2X’ (= omy ocd ss 
: x Ya . Y1 ae Wai = x’ my 
Hence, on account of (1), we have the invariant functions 
/ 
Vk yin (x) eS 
] ? 
? 
Yi Van Yi 
from which it is only necessary to eliminate x’ in order to obtain our required 
differential invariants: 
9. The general projective group in one variable 
a ¥4) ¥74 
y® == (4) y?) — y) 
y—y® * y—y® 
Using t, r,s as auxiliary variables, R takes the form, for ultimately coinci- 
dent points 
leaves invariant x and R= 
Be bt ...), 
