131 
The remaining equations expressed in these new variables are 
11 ae Sau iy gr a 
ae vise (01) . Fann, ) 
te (1) 
W bf 2 4 Si (ay, S=0, =1....r—5), 
dl, 4 dQ 
n 
o 3 hi 
Y'f ——— >i Sie Fen —— 
D) iF) 
The determinants D, (s—0,1.... n—r-+3), of the matrix formed from 
equations (1) as in 21 will be solutions of (1). Ds will be linear in ¢, but Y'£— 0 
requires the ratios of (’s. We may write our invariants as 
o) = =2 >; ey 18; = ID): 6 Dig (Secon ty HET coon WP apa) 
Ky 1 XS X38 
23. p, 2xp-+ yq, X?p +-xyq- 
This projective group, leaving invariant the x-axis, furnishes us the com- 
plete system 
n . n n : 
ae =i {2 nua i \= ai {xe Gh + xiyi EN Sasi 
1 il ay i 
Ades dxi | > dyi J ” axi ly: J 
The first of these equations has solutions yi and 4; = x, — xj. The last 
equation then becomes 
n 
ye APE df } 0 
= l j dj sain YiPi dy; ’ 
with solutions 
1 i 63 
UW = a == Ox (= ; De Vel 
ghey dt = df 
ae Sel epee SOUS: — 
7 dy, a ee v3 dv; z aa eda 
whose solutions we may choose in the forms 
ul ee eG 
ete eee — a. = EOS Wes (Gas ean kei... Bie 
Our inyariants are 
xX, — Xk k eng b b-€5) —— 2.8 
— : Pe te I ee, C= es, 2, 
25 2.55 Sh 2. Gi 9S) NG: YiYe 255 ee SEVP ne! 
whose geometric significance is apparent. 
24. | yq, p, Xp, X7p + nyq |. 
