123 
(C.) The families of curves x = constant, y = constant, remain invariant. 
5. lq, yq |. 
The complete system corresponding to this group, 
ede ey ats 
Ef ayiee ‘ dyzt a? 
has as solutions 
xX, and i — (yi— yx): (yi — yo), (11)... Dy kK==3).... 0). 
Hence. xi and x are the invariants. 
6. |q, ya, y’q). 
This is the general projective group in one variable, and leaves invariant xi 
and the cross-ratios of any four ordinates. 
yi dyi == \()} 
n n n 
ahi Sys pea 
_ vd . — 
tent eaane 5 
The first two equations of this system have solutions xi, {x of 5 above. In- 
troducing these solutions as new variables in the last equation, we have 
n 
= dt 
>k Vi (We —1) did, Ue 
3 k 
whose solutions are 
—— jl We, — == 
ee Ws betiay yl Sal yi (Ae 
2.6 Gil : = : ; 
i Ws Ye gt = Ye 
5 30))p 
7. {a ya pl. 
This group leaves invariant 
tie —'(¥, — yu): (¥1 —y2), and ¢4;=x, — x, (k=3 ....n, j=2....n). 
8. 
q, Ya, ¥2q, Pl. 
The invariants of this group are clearly 
rs 7277, yu= 7 as in 6, and 
Yo hee Vite ary ture 068 
oj = x, — Xj, as in 7. 
J" |g, pe xpem evan | 
The solutions ~j = yi — yi, ¢i — x1 — xj of the first two equations obtained 
from this group, when introduced in the last one, give 
n 
df df 
ee ey 1 0. 
at \° ddj 4e Cc Uj nt 
