102 Proceedings of Royal Society of Edinburgh. [sess. 
The mode of formation, seen to hold in these two cases, being 
then supposed to hold for 
(a*, 4 , . . O 0 ”- IkX 2 • • • 
is attempted to be shown to hold for 
<X 4 . . a;;., A :r- !«. . . 
that is to say, the case for n variables, A v . , A n is sought to 
be made dependent on the case for n - 1 variables, A x , . . A n _i, 
The process followed is to change 
p remaining the same in both, 
the first factor into 
(4 4 • • •> a * 
\ h 
V ^ z+1 ’ 
\ h 
"•o J 
^■z-V ^z+V 
. , a 
express the second factor — the alternant — in terms of n alter- 
nants of the ( n - l) th order, and then perform the required multi- 
plication and condense the result. This being satisfactorily 
accomplished, it would not of course follow from the two special 
cases previously dealt with that the theorem had been established 
in all its generality, but merely that it held for any number of 
variables A 1} A 2 , ... so long as p was not greater than 3. The 
passage from one value of p to the next higher — which is left 
unattempted by Schweins — is not free from difficulty, as will be 
seen on trying a particular instance, — say the passage from 
\a r b s c t d u \ . (a h b h + a h c h + a h d h + b h c h + b h d h + c h d h ) 
to 
\^l s c t d u \ . ( a h b h c h + a h b h d h + a h c ll d n + b h c ll d l1 ). 
Several special cases of the general theorem are noted, where a 
number of the alternants on the right vanish and where con- 
sequently a comparatively simple result is attained. 
The first of these is where the indices of the alternant to be 
multiplied proceed by a common difference h : the identity then is 
a a 
j a 2 ’ 
• , o 
h\(p) 
. a+h . a+2h 
Aj A 2 
a+nh 
■I . ci-\-h . u-j-2 h 
-IAj A 2 
. a+(n-p)h , a+(n -p+2)h 
A n-p A n-p+1 
A w I \ “ I 
) 
)h y 
L n 
The second is where h= - h , and the indices proceed by a 
common difference ft, the result then being 
