554 Proceedings of the Royal Society of Edinburgh. [Sess. 
The following are easy to prove, 
Ka = (- p- W, zs/3 = (- TZ-$ n r = l S n r.us ... (3) 
Pass all the ?tf’s of (1) to the end and then cancel us n . In this process 
the left S w becomes n S w , the right S„ becomes n _ c S n , S c and S w _ c become C S C 
and n _ c S n _ c respectively. Also the left is multiplied 1) times and 
the right Jc(c-f- l)-f c)(n — c + 1) times by ( — ) n ~\ Hence 
= 2 2 ( ~ ) c(n - 1) . c S c /3 (c, . n _ c S w fe.n- c S ?l _ c d (n - c) ) . . (4) 
This may be modified in several simple ways. The following statements 
may be proved by the reader : 
V n - l q = TZqTZ~ l (5) 
( _ ) c ( w -D. c S c /3 (c) = P n-1 c S c /3 (c) = £r c S c /3 (c, .C7 _1 ... (6) 
The following may be taken as the standard form of (4), 
- ^ Afe. c S^) (7) 
It may be obtained thus : Multiply each side of (4) into n S~ l /3 {c) /3 (n ~ c) so as 
to be free as to the sequence of factors in f3 {c) and /3 {n ~ c) ; transpose f3 {c} 
and /3 {n ~ c) ; change c to n — c. 
We now turn to complementary proplacements. Let every fictit l be 
replaced by usi where us is a product of all the fictits in any sequence. On 
account of (5) it will be seen that with us thus to a certain extent arbitrary 
uSi can be made any one of the complements of i and we will at any stage 
suppose us to be replaced by some other such product, for instance by us~ x 
or by rus. This substitution of usi for i obviously constitutes a pro- 
placement for which Law A as well as Laws (1) to (4) are retained, for 
ZZl v ZSl 2 = - 7Zl 2 .USl v (GT t] ) 2 = ( - ) n_1 G7 2 1 2 = (ZSl 2 ) 2 = (8) 
Thus we may put 
Rtj = zSi v Ri 2 = zSl 2j . . . . <9) 
where R is a proplacement in our technical sense. 
If Vc ~ l'l" .... l ( c) it is easy to find an expression for Ru c = uSi uSl" .... usi {c) > 
for every transposition of us with an l requires us to multiply by ( — ) n-1 . 
Hence 
Rv c =(-)^- 1 ^- 1 to c -u c , 
or if q c is S c q 
Eq c = (- Dus c q c (10) 
Remember that for different forms of us, us 2 has but one definite meaning. 
