I 
40 Proceedings of the Royal Society of Edinburgh. [Sess* 
Hence the above ratio tends to the limit 
(1) • 
m 
m - n 
m m—n 
n m — n BC n 
VI 
A /l 
n 
for indefinitely large values of r, which limiting value must therefore be < 1. 
If this condition be fulfilled, then there are n roots of this trinomial form 
belonging to this operand and the operator D^ 1 . D c > n . D A ", and 
CM 
vice versa. 
Similarly for the roots belonging to and the operator 
- 1 -(A-A 5 
T) T4 T) m 
^ A • U C • 
in the same equation A^'drB t r ,J 'drC = 0, and given by the formula 
/C\i 1 A C 
X = ( — p -I • • — 
VB/ m 1 
n—m+i 2n—2m+l 
1 2n - m + 1 A 2 C m 
n + 1 
+ 
B 
m 
m 
9 | 2ft +1 
A TO - 
3 ft— 3m +1 
I 3 n — m+ 1 3 n - 2 m +1 A 3 C m 
+ - * ~ - • ‘ -Oi 3ZM- » etC -r 
II b 
m 
m 
3 I 3ft+l 
A m 
we find the test for convergency to be 
( 2 ) 
n \— n - m C TO . A n 
m . . — <1. 
n - mj m 
And finally for the roots belonging to 
n 
B™ 
B 
A 
and the operator 
-1 
] V . Tv ft — m T T n—m 
J -'(— c) -^a - • 1; 
given by the formula 
?•/?. — 1 
n 
2m— 1 
+ m- 1 C 2 A^ 
'p\ft-m J Q J^n—m 
A/ n — m 1 1 1 n — m n — m 2! ^—2 
B r 
B 
3m— 1 
- 7/1 
1 2n + m - 1 n + 2m - 1 C 3 A. n ~ r 
n — m 3 ! 3,1-1 ’ e ^ c -> 
n — m n — m 
B 5 ”-’ 
we shall find the test to be 
(3) .... 
m 
m C . A n ~ 7 
n J n — m 
n 
m 
<1. 
m 
^Raise the factor corresponding to L here to the power of 
