166 
where S denotes the sum of all the differently arranged pro- 
ducts of m+n factors, of which m-j are ys, n—v are Z8, 
are js, and vy are ks. Now the number of these arrangements 
in S§ is 
(m+n)! 
(m—p)! a=v) lv? 
and. § itself will obviously be of the form NZ (u, v), N being 
some numerical coefficient depending upon m, n, m, and v. 
But as the number of differently arranged terms in 3 (p, v) is 
(u+v)! 
piv! ? 
it is plain that we shall have 
(m+n)! 
N= 
(m—p)!(n—-v)! (u+v)? 
and consequently, 
aalval 
C,= aon = : 
(m=)! (n=v)! (ut v)! (us ») 
«Thus, we have found that C =C,, and as this is true for 
the numerical coefficient of every term in the development of 
7 y™2", we are warranted in concluding that this latter ex- 
pression is equal to the 
((m+n)!)* 
m!n! 
(m+n)! 
m! n! 
part of the sum of all the 
differently arranged products of the m+n factors, of which m 
are equal to y+ ja, and n to 2+ ke. 
‘¢'The statement ofthis theorem, and of other similar ones, 
may be rendered simpler and more elegant by our assigning a 
name and symbol to the last-mentioned expression. I propose 
to call.it the mean value of the product of the factors combined 
in different orders: and for the present to denote it by the 
symbol 
M(y+ju, 2+ kn). 
We may now proceed to interpret the expression 
ee a rt 
