126 
formed by filling the (r + 1)-th column successively by the (r +1), the (r+ 2), 
.., the n-th column, The invariants are clearly xi and Ds, (i= 1.... n, 
Sl, e.~ UT); 
15, | Xia X2q. Xoq, ---. Xraq, ya 
| aa 
This group furnishes the complete system 
W (ae he (xi) 1 Seer == >; Vi dt —— 
k —— 5 k 1 . dyi —_— , =—= a a l. dyi — . 
The solutions of Wxf—0 are clearly xi and the determinants Dz, (s—0,1... 
n—r), of a matrix (Mr—1) constructed similar to (Mr) in 14. Yf—0O requires 
the ratios yi: yx to appear in the final solutions. Hence, we may write as invari- 
ants x; and 
fe— DE (t = ee —) 
| 
e%Fg, xe%=Xq, x"e%Xrg, .... xke™*q, p 
16. m 
Kooi em, 2k ¢,+m=r—l,r>2 
From this group we obtain 
yr . ue di z) = dt ¥ 
\W kt = Si (xi)tk eMXi, == "0; KES SiS = Oth == 0 te eee 
1 dyi 1 dxi 
The solutions o;=xj— x, of the last equation are also solutions of the 
system. By dividing the remaining equations, respectively, by e'K*1 the expo- 
nents of e bacome functions of 9. The independent determinants D., (s=0, 
1... n—r), of the matrix (M;,— ,), formed as indicated in 15, will be solutions. 
The invariants are, therefore, 9; and D.. 
> - 
| CEQ, KONG sk Cg. < .s RC ap 
a7 24) ee ‘ 
| k=1....m, x¢é,+m—r—2,r>3 
1 
The complete system given by this group is 
D n q n 
Wt f= di (x1)*. oo z = 9 YT Fi On BG fae ee 
k 1 dyi 1 ayi 1 dxi 
i a ae NE ea) 
