428 Proceedings of the Royal Society of Edinburgh. [Sess. 
and Euler’s theorem regarding the differentiating of homogeneous functions 
giving 
there is obtained 
B = 
m - 1 
mu 
= x 0 u 
0 ~l~ x^Uy "H 
+ x r u r , 
(m — 1)« 0 
= X 0 U 
oo + ^j^oi + 
-t x, r u^ r , 
(m - 1)^ 
= X 0 U 
10 + X-yU-yy + 
4 - X r U\ r , 
(m - 1 )u r 
= X 0 lt 
rO + X\U r \ + 
+ x r u rr , 
XqUq — mu 
U 1 
u 2 ... 
u r 
X 0 U 10 
U n 
U u • • • 
U\ r 
X 0 U 20 
U 21 
U 22 ... 
n ‘lr 
XqU t0 
U r 1 
u r2 ... 
U Tr 
*4 
U 12 ' ' 
. . U lr 
u 0 
U Y 
. U r 
- m 
— u 
u 2 1 
U 22 • ' 
, . U 2r 
+ x 0 
u 10 
U U 
■ w lr 
m— 1 
U r l 
U r 2 . . 
. u rr 
m — 1 
U 20 
u 21 
. u 2r 
u r0 
U r 1 
u rr 
By similar treatment, however, the second determinant on the right of this 
1 
m - 1 
m — 1 
(m - 
- IK 
(in - 
-IK • 
. . (in - 
- 1 )u r 
U 10 
“n • 
Ui r 
U 20 
U 2y 
u 2r 
U r0 
U r l • 
u rr 
M oo 
u oi • 
M 0 r 
U Y0 
Uyy • 
Wlr 
u r o 
u r 1 
Uyp 
> 
m 
uR 
. + 
2 
H ,, 
m - 
(w - 
i) 2 r+i 
so that finally we have 
B 
as desired. 
As a corollary it is noted that if H r+1 vanishes identically, then 
(m - 1)B + muR r = 0 , 
mu 
Uy . . 
. . u r 
(m - 1 )uy 
Uyy . . 
■ . u lr 
i.e. 
(m - 1 )u 2 
U 2 y 
(m — 1 )u r 
U rl . 
. . u„ 
