130 
The remaining equations W f may be expressed in terms of «, ¢ in the follow- 
ing forms: 
df df 
fig aac Tos 
+3 PA (Gi) so 
i Hes 
og = 0 [d=3 —4)} 
Phare (a) a, Fo, (eet Ses hy 
Sen oF 
The solutions of these equations may be expressed as the determinants Ds, 
(s=1, ....n—r-+ 3), of the matrix 
ama A ahem TEM NTE 
n 
ea = iil —id 
1 Oy O;_345 n 
—d 
Gil fad Geis: Ahiacl- spk aecwareielars 
ache Ge ee) ete W cae eR) Fe Lee) were) ie! tule (6) te, 
Seem, ee ae Oe Ww OY ere 6. 18) 6 4G) eye) Be me wee. Ve Ke) lau @ Pie) ie 
The required invariants are, therefore, 
2.) > SI Xo 
0) =>— : 
mesh *) Ds, (I= 4. nj s=1) 26 m2 3) 
X,—Xl X,—x 
29 GyexGh xed dneni =a yq, Pp, Xp, x*p + (r— 5) xyq 
c r>o 
7 di sis dt 
W,f — Di xit — = 0, (t= 0, 1.... r—5), Vis — > 
dyi ayi 
1 1 
= es i re o df 
xX ——— = ax ==(), A, i — om 1 Xi AES — 0, 
Ser ie es te 
= = ae iad (r— i») XiVi dy; ‘ 
The solutions of Wof =0 are Wj—y, those of X,f=0, Xfi=0 are 
Ux = (x, — xx): (x, — x,.). Xf expressed in 7, u is 
n * . 
aie ae Yt Re aie df 
1. f == a | Uk (uk 1) dux af (x 5) Uk. dw sr (r == 0)! 9) Wo Z = 0, 
whose solutions are 
us (u)—1) - Wik -. 
SS eee SKS FO es Se 
Ti (lls al) (ux —1)t-® 
vw ( = ‘ae (l=4...n,k=3...n). 
us — 1 
