1893-99.] Dr Muir on a Single Term of a Determinant. 461 
(24) The reason of this unlooked-for relationship becomes more 
apparent when we actually try to form in order the combinations 
referred to. For the first three cases the results may be tabulated 
as follows : — 
Things 
i given for Combination. 
a. 
a, lb. 
a,bb,ccc. 
(0) 
Sets of 1 letter, 
a. 
a,b. 
a,b,c. 
,, 2 letters, 
ab,bb. 
ab,bb : ac,bc : cc. 
,, 3 letters, 
abb. 
abb : abc,bbc : acc,bcc : ccc. 
,, 4 letters, 
able : abcc,bbcc : accc,bccc. 
,, 5 letters, 
abbcc : abccc, bbccc. 
,, 6 letters, 
abbccc. 
Now it will be found that the combinations in any column here 
are got from those of the preceding column in a manner closely 
resembling that specified in the practical rule for the construction 
of the table of § 16. For example, if we wish to form the com- 
binations of four letters taken from a,bb,ccc, we go to the preceding 
column and take over without change the combinations there 
given ( i.e ., none) of four letters, then take the combinations of 
three letters, viz., abb, and annex a c , then the combinations of two 
letters, viz. ab,bb, and annex two c’s to each, and finally the 
separate letters a,b, and annex three c’s to each. The correspond- 
ing operation in connection with the table of § 16 is 
V 4i4 = 0 + 1 + 2 + 2 = 5. 
It is thus seen that if the problem be to find not the combinations 
themselves but the number of them, the difference-equation is 
exactly the same as before, and the initial conditions are also the 
same if we make the usual convention that in every case when we 
omit all the letters one combination is to be counted ; that is to say, 
r — i 
^l+2+. . . + (W-1),0 
