470 Proceedings of Royal Society of Edinhurgh. [july 16 , 
(3) When the indices in a term of % are of the most general 
character, the value of K can be found by a repeated double 
application of the principle of Mathematical Induction. [We shall 
find it convenient to denote throughout by S the corresponding 
complete function involving a in any case.] 
First, to find K when two indices are alike and any number 
diiferent, we have when two are alike and one different, 
A = 'W^c^d^ = S - - a^tlTc^ ; 
13 
thus ^ = 0-1-12 = - 
To deduce for case when two indices are different, we have 
A = %h^c^d^e^ 
and thus 
= S - - a^th^cHP - a^%ired ^ ; 
\i 
Now assume this law for case of two indices alike and any 
number /x different, where 
A = Si"‘e” . . . g‘k>k>, and ( - lf+2. 
then, when fi + 1 indices are different where 
A = . . . hW , 
we have, expanding in the same manner as before, 
K = + 1^+2 
=(-lr+^ 1^2 (^+i) =(-if+s. k±i. 
Thus, if N he the whole number of indices of which two are 
alike and the rest unlike, w^e have 
^=(- 1 )".^ ( 2 ). 
In precisely the same way, starting with three indices alike and 
one different and using preceding results, we deduce that, when 
three of the N indices are alike and the rest different, 
( 3 ). 
