OF THE COMPOSITIONS OF NUMBERS, 
877 
hence the numerator is free from such a term and every term in it contains one of the 
quantities 
; 
but the whole fraction contains no quantities 
and thus it is manifest that the fraction can contribute no term which is a function of 
50. To simplify the discussion of the remaining fractions it is necessary to consider 
some particular properties of the typical numerator. 
The function 
P ... 
K1K2 • * • Ki 
may be expressed as 
.( 2 ) 
(t-1) 
or as 
{t (2^ — 2) a, + 2aJ . . . + 2a,) 
— t {2^ —2) a,a,a,^ {A,^ + 2a,) . . . {A,^ + 2a,) 
+ . . . 
+ ( —y (2^ — 2) a,a,^ . . . ... 5,^ 
the summations being in respect of the t numbers 
^13 ^2> 
Writing 
we have now the identity 
#^ 1^2 . . . ^ KiK.2 , , . K ^2 • • • 
= S (23 — 2) a,a,A,^A,^ . . . A,^-\-t (2^ - 2) a,a,a,A,^ . ■ . A,^ 
+ . . . + (2^ — 2) a,a„^ ... a. 
51. To establish this it is sufficient to observe that the coefficient of 
