1888.] Professor Anglin on Alternants. 471 
^7ow assume this law for case of p indices alike and the rest 
different, where 
(4). 
Iz 
3. To deduce the value of K when p + 1 indices are alike, we have 
Avhen p + 1 are alike and one different, 
A = 'Zh'^c^d^. . . 
= S - ... - ; 
thus 
— / _ 1 \ p-\-^ 
' TE±r 
Assume for + 1 indices alike and any number /x different, in 
which case 
_ / _ 1 \ ja+i3+l I /X + + 1 
^ ‘ 1^+1 ^ 
then if /X + 1 be different we have 
= ( _ i\i*+i>+z . + ^ / /t+1 j\ 
\P \P + 1 / 
-C_l'\<»+i’+2 \p+P+'^ . 
'■ > ■ |j>+l ’ 
which thus establishes the law for y indices alike and the rest 
unlike. 
4. The case of p indices equal to a, two equal to yS, and the rest 
unlike is next deduced in a similar manner, when we get 
A=(-ir. 
iA 
|J>|2 ’ 
and more generally when p indices are equal to a, r/ equal to and 
the rest unlike, when we shall find 
A'= (-!)«. 
J£_ 
\p\<l 
