124 
The required invariants of the group may now be chosen as 
ee ee 
X1— X2 yl— y2 yl—ye2 
(k=3....n). 
10. q, YG, P, XP 
Comparing this group with 7, we have at once the invariants 
ig ey ae (k= t3 ean) 
RAO Bye ANT = FF 
11. | q,yq, y7q, Pp, Xp 
By comparison with 8 and 10, it will be seen that this five-parameter group 
leaves invariant 
= 
Gi 2S re 
y we we, Ur = — 
VA Sale eile mame x) 
Lore) AG eee dak Sr Bae 
Comparing with 6, it will be seen that this group leaves invariant the cross- 
ratios of any four abscissas, and ordinates: 
ae Marca 2 Neath 
sl] = 
2S 2 Ges 
— : — — eH beset pia) 
y2—ys yi — ys x2— Xs, Xj — X8 
Bee ol) ay es Dt ele 
This group furnishes the complete system 
=— S {x a + yi = fa Si {xe 2 +ye 5} ==\(), 
A ee Ls) 
ri dxi dyi 
pans gj=x1— xX}, W=y1— yi, >= 1 — y1 
as solutions of the first equation, we have the remaining equations in the form 
n Fy 
ee ee Le Wms 
W, me ke Trak ee? a 
n . 5 
aS 2 at wo ot at __ 
W.f —— J " oj doy + Wj dij \ =F 0 do —— 0. 
Solutions of Wit 0 may be taken in the form 
o Ik bo 2 
Yh me i — Y ee — Bice — Lk 
Qo Wo o 
Expressing Wef — 0 in terms of these new variables, 
= df df ». df df 
2k {um (1 — ux) se t ve(1—vx) 5 +- w1 (1 — 01) Ae ee ae 
