VOIy, 6, 1920 
MATHEMATICS: E. B. STOUFFER 
647 
From (2) it follows that the most general transformation of form (1) 
which leaves {B) in the semi-canonical form is given by 
yk 
X=l 
where a^^x ^-re arbitrary constants. Equation (4) shows that such a 
transformation converts (B) into a new system whose coefficients wiki 
and their derivatives Tri?; are given by the equations 
n n 
D 
tSi =22 ^^*'^mVm*. 0 = 0,1 m- 2), (9) 
X=l M=l 
where is the algebraic minor of in 
If the transformation (8) is made infinitesimal, it is found that all 
seminvariants in their semi-canonical form must satisfy the system of 
partial differential equations 
2 2k^''s^ 
0, 
k=i 
(r,s = l,2. . .w;/ = 0,l,2. . .,m - 2). (10) 
For r = 0, / = m — 2, there are n solutions given by^ 
r! ^ \&x,,„_2/ 
''^\,\,m — 2 '''"1,2, m — 2- • • -TTl, n, m — 2 
'^2,1, m — 2 ^2,2, w — 2- • • •7r2, M, OT — 2 
'^n,\,m — 2 "^n, 2, w — 2 • • • •'^n. n, m — 2| 
(r = 0,1,..., n - 1) 
For r=l,2,/ = m — 2, solutions are given by^ 
n n 
1=1 i=i 
^,2 2(-— ,-^)'^'^^"^ 
(f + ^ < ^ = 1,2,3; t < s). 
For T = 0, / = 0, 1, . . . . ,m — 3, solutions for each value of / are given 
by 
J,'"" 
1 ^ / a V (r Jr s <n;t= 1,2; i < sA. 
