154 
This equation requires x, y to enter in the final solutions to the degree zero. 
Hence, we may write at once the invariants in the form 
Di 
Jp p= 1131) 21128], (l= 4... m). 
I and J show that by the general linear group the ratio of areas remains constant. 
27. The general proejctive group 
P, qd, X4, XP — Yq, YP, XP + Yq, X°p + Xyq, Xyp+y’q |. 
The members of this group extended and equated to zero furnish a complete 
system of eight linear partial differential equations, the first six of which are 
identical with those of the general linear group, and therefore have solutions 
I, J, defined in 26. The last two equations, 
a r : n 
gt RS an iri LSS BO pe 
Bi fat tg Ja at alae 
when expressed in terms of I, J, become somewhat complex, 'viz.: 
df 
if - 
Q) Ty Gy —“Ie—V) Gp + Sm | Tw CFn Ind Fe $F) gt 
Jm (Le Jm— In Jig —Jm—+ i :;—1) aa=0 
n 
I, (J, —1I, +1) = -+3m4 Jim (1, Im— In’, —ImHT) eh 
5 m 
9 
(2) ( 
Im (I, Jm—ImJ, —In+J,+1 — Loo. 
After considerable manipulation, the solutions of (1) are found to be 
zs m a 
Im J4 l, __ 9m I I (Is ? 1) (Mm —Oies.5 1) 
Sra ORE 5 gest 77 (Ou — W421} 
With Iy, om, vm as new variables, equation (2) becomes 
df 
May, an + ime = dom 
= 0, 
with solutions 
Qn = — — Wm 
