104 
Proceedings of Royal Society of Edinburgh. 
[SE£ 
and corresponding to the identity 
Qn,r = ^n— 1 , r d* r_i 
we have 
^n,r ~ ^n—l , r d" h ?i _2 r—1 • 
Similarly, since the proposition of § 3 may be written 
^n,r ~~ r r 
we have the identities 
~^n,r = l^-J(?i+3}-) , (n—r, j 
■K-w,r Kji-2, r d* i, j»_i • 
(5) In the proposition of § 2 the condition imposed was such that 
the selection of any particular element to be one of a set meant 
the rejection of the element immediately following it in the 
original row. A manifest extension of this condition at once 
suggests itself, viz., that the selection of any particular element 
shall exclude the k elements immediately following. The more 
general proposition thus obtained — and obtained in exactly the 
same way and almost as easily — is 
If C n k r denote the number of combinations of n elements taken r 
at a time subject to the condition that no element is to be taken 
along with any one of the k elements immediately folloiving it in the 
original roiv, then 
^n,r = ^n-k(r+ 1 ) , r • 
Derived from this in the same way as the proposition of § 3 was 
derived from that of § 2 we have the second general proposition — 
If K k r denote the number of different sets of r things obtainable 
from a row of n things by leaving out of n - r of them which form 
(n - r)/k k -ads of consecutive things , then 
Ki = C|(n +& Mr- 
Of course each of these may be utilised as the less general pro- 
positions were dealt with in § 4. 
( Issued separately April 1 , 1902 .) 
