1901-2.] Dr T. Muir: Note on Selected Combinations. 103 
true for the next case, vis., where r = s + 1. For we shall get the 
combinations desired by taking 
a 1 along with s things taken from a B , a 4 , ... , a n 
a% . . . ... ... ... $4 , • • • J &n 
the total number of them thus being 
C n — 2 , s "1" ^n— 3 , s 
which by hypothesis is equal to 
^n—s—l, s d" C'n— s— 2, s 
and consequently equal to 
c„ 
a.,* 
u — (s+l)+l, s+1 
as was to be shown. 
(3) As a second instance let us take the condition that each set 
of r things is to be obtainable by deleting from the n things a set 
of n - r of them so chosen that they form J(w - r) pairs of con- 
secutive things. 
The full list of pairs of consecutive things is 
(«i » ® 2 ) » ( a 2 » a z) > ( a 3 > 
K 
As each of these n— 1 pairs has an element in common with the 
pair which follows, we cannot in selecting \{n - r) pairs for deletion 
include among them two consecutive pairs. It follows therefore 
that if the desired number of sets of r things be denoted by 
K' w , we have 
~^-n,r C n _ 1 1 1 (n—r) , 
C n — 1— i(n— r)+l, i(n—r)t "^y § 2 , 
, \(n—r) > 
(n+r),r* 
(4) The proposition of § 2 may also be written 
P — C 
KJ n+r— 1 , r ) 
and we thus see that any known identity connecting several C’s 
has corresponding to it an identity connecting as many C'’s. For 
example, corresponding to the identity 
C n>r — C K t n-r 
C n ,r ~ C‘2n— 3r+l , w— 2r+l ; 
we have 
