Dr T. Muir on the Theory of Determinants. 
523 
Z = —a )• — a )... — a) . — a^) . - a' ) . . . — a } 
<hip!" — a!') . 4>(a'" — — a") . — a ") . — a"). . — a'^) 
<^(a(«)-aO*-D). (ill. 22.) 
And it is this form which Scherk seeks to establish. The mode of 
proof is again the so-called inductive mode. In the case of heo 
permutahle indices the law is readily seen to hold. We thus have, 
preparatory for the next case, 
/2 k k\ 12 12 
P( a ; a , a ) = ^(2 -\) a a -f <^(1 - 2) a a , 
\2 h h/ 12 2 1 
( 2 3 1 \ 3 2 3 2 
a\ a , a) = <^(2 -1) aa -l- <ji{l -2) a a, 
2 h h' 12 2 1 
/23 2 \ 13 13 
P( a ; a , ct) = <^(2 -1) aa + ^(1 - 2) a a . 
^2 h la 12 2 1 
But “nach dem Obigem ” 
k 
k\ 
1 
^2 
3 
1\ 
2 , 
^2 
3 2^ 
3 , 
^2 
k 
k\ 
ia ; 
a , 
,a) 
1 = - aP( 
a) 
- nP 
a ; 
a, «) 
-h «P 
a : 
a , 
a . 
\3 
ft 
iJ 
3 ' 
^2 
h 
iJ 
3 ' 
^^2 
h iJ 
3 ' 
^2 
h 
h ' 
Consequently 
/S k k\ 1 ( 3 2 3 2 I 
P(a; a f a)= -a\ </>(2 - l)a a + <^(1 - 2)« a ? 
\3 /« /j/ 3 ' 12 2 1^ 
2 ( 13 1 3 I 
-a \ (f){2 - \)a a + cj){l - 2)a a f 
3 1 12 2 1'' 
3 r 12 12) 
-a\ cji{2 - \)a a -t- cf>{l - 2)a a !■ • 
3 ' 12 2 1^ 
But as -<^(2-1) = ^(1-2) 
and - 1 = cf>{2 - 3) 
and - 1 ^ <^(1 - 3) , 
w^e may substitute 
<^(1 - 2)</)(2 - 3)<^(1 - 3) for -<^{2-l). 
