1899-1900. J Dr Muir on the Theory of Alternants. 107 
II . a+h . a+2h 
K a 2 . 
. a+ip - 1 )h a +(p+l)h 
• ’ * A p-1 A p 
a+nh a+(n+m)h ' 
• • 1 ^n -1 A n 
) 
|i . a+h a+2h 
| Aj A, 
= (4.4 
A" 
’’ 
i « < 
i i 
IS; 
•> a:j 
|(m) 
- (a; 1 , . . . 
A 71 
'» A n 
.. 4] 
|(m+l)’ 
+ (a?, 4 . . . 
A h 
’» ^71 
T~ p - 2) . 1 4 4 , . . 
•,4] 
| (m+ 2 ) 
(-)’‘A4X • 
• *) 
A h J .[44 •• 
- 4] 
| m+n 
Again starting from the same initial identity we obtain the 
analogous series 
Mp jS 
+ 
+ 
i 
525 
II 
4 
Ag-h 
— ]\lp_i )g _/ t 
- 
Mp_2,s-2/i — ~ -^w-jj+2 < 
• A S -2 h 
+ Mp - 2,s - 2h 
+ 
Mp_3 jg _3A = +lSr n _p-)-3. 
■ Aj,_3 h 
(-)p + 0 = ( - ) p 1 N n . A s-ph) 
and by addition have 
Mp )S = KA-p-fi . A s -h ** N w -p-|-2 i A s _2 h + • « • • ( — )P~ 1 N n • A s -ph 
or 
II .a+h.a+2h a+(jp-l)/i a+(p+l)h a+nh 
| A 1 A 2 • * * * p- 1 A p A n-1 
A[) 
CM 
rH 
II 
A h 
\(»-J>+l) II ^+ h \a+2h 
) ‘ 1 A 1 2 • * • 
. a+(n - l)h . s - hj\ 
■ A ?i -1 A n ) 
- (4 , 4 • • 
A 74 
71 
~yn-p+2) ||^a+^a+ 2 /i 
a+(7i - Vjh s - 2A\ 
' A n- 1 n ) 
+ (a'*, 4 , • • 
A 74 
■ n 
^(n-^+3) |^a+A^a+2^. 
. a+(n -l)h s- 3 h\ 
' n- 1 A n ) 
(-4(4 4 , 
• • • 
, A") (n) . |iA“ +, ‘A* + “ . . . 
. a+(7i - 1 )h s -ph\ 
* A w-1 A n ) 
so that by substituting as above for each of the alternants on the 
right and dividing both sides by |^ + ^^ +2/i . . . A^ +7?/t ) there 
results the alternative theorem 
