524 
Proceedings of the Royal Society of Edinburgh. [Sess. 
A form of (7) which is often useful is 
^n—cQ ^ c* v£L ~ ® cQT n'^ n ? n^c^n Q_ n *^*« • • • (®) 
* We now proceed to prove the theorem numbered (12) below. 
[In what immediately follows the reader will probably be assisted by 
mentally naming zs c “ the common product ” ; to what two expressions it 
is common is obvious enough.] 
Assuming that f3 v ...., /3 b , y v ... . y c are independent we may put 
n = b + c and r n = S b+C zz b m c - Thus we have 
S &+c tiT 6 tzr c .S c trT c = S 6 (S &+c tn' & tD' c .^iT c ) = S 6 (S 6+c CT & CT c .S c CT c ) . . . (9) 
[The middle form is given directly by (7); the third form by noticing that 
since the first is a 6 th part, it must be the b th part of itself.] Although we 
assumed the b + c fictors V5 b xs c to be independent, (9) is clearly true also 
when they are not independent, since S 6+c - 57 & tt 7 c which occurs in each form 
is then zero. 
Changing w b in (9) to m a w b it follows that the first two of the following 
three equations are true universally, 
^a+b+c ^ \ 
= S a +j,(S a+6+c CT c CT a trr 6 . fe c CT c ) /■ (10) 
= Sa-f&(S a + c ^ c ^ a . S^_)_ c ttT & W c .) ' 
We proceed to show that the third and fourth of these forms are universally 
equal to one another. 
First suppose the <x + b + c fictors Z3 a m b w c are independent. Let a, a 
be two of cq . . . . a a ; /5, /3' two of /3 1 .... /3 b ', y, y' two of y x ... . y c . 
Then linear variations of the following types 
y to y + xy, 
a to a + Xy, Or to a + Xa! , 
j3 to /3 + x/3 or to /3 + xy , or to /3 + xa , 
do not alter the value of either the third or fourth forms of (10). All 
these statements are obvious from the properties of combinatorial parts 
except the last, namely, that changing /3 to /3 + xa causes no change in the 
value of S 0+6 (S 0+c w c tiy a .S 6+c t3r 6 izr c ) = g (say). Variation from /3 X to /3 1 + ^a 1 
adds to q 
( ) h+ ° 1 ^Sa+&(^a+c^c a l^a-l"S&+ c ^&-l^c a l) ( sa y) 
* I find (April 1908) that (12) below may be much more simply proved thus : The 
“ transference ” of rs x may he effected in x steps by transferring each fictor | l5 | 2 • • • • £» of 
vs x in succession ; and we have to prove the theorem for one such transference (say of £ x ) 
only. The last is quite simply proved by expressing all the fictors in fictits c , which 
constitute the fictorplex rx c . [When the fictors tx c are not independent the theorem 
obviously takes the form 0 = 0.] 
