104 Proceedings of Royal Society of Edinburgh. [sess. 
Each result is seen, as in the preceding case, to be a sum of 
alternants differing only in the indices from the alternant which 
is the subject of multiplication. Further, it is observed that this 
difference is a difference in excess, the indices of the multiplicand 
appearing in all the terms of the product, so that the only 
difficulty is to ascertain what addendum is to be made to each. 
The next observation is that the addendum is a multiple of h, and 
that in the three cases the multiples are the following : — 
2 h, 
Oh 
2 h, 
Oh, 
Oh 
3 h, 
Oh, 
Oh 
1 h, 
1 h 
Oh, 
2 h, 
Oh 
Oh, 
3 h, 
Oh 
Oh, 
2 h 
Oh, 
Oh, 
2 h 
Oh, 
Oh, 
3 h 
1 h, 
1 h, 
Oh 
2 h, 
1 h, 
Oh 
111, 
Oh, 
Ih 
2 h, 
Oh, 
1 li 
Oh, 
1 h, 
\h 
Oh, 
2 h, 
l/i 
1 h, 
2 h. 
Oh 
111, 
Oh, 
2 h 
Oh, 
Ih, 
2 h 
111, 
lh, 
lh 
The law of formation seen by Schweins in these coefficients of h 
is to be gathered from the sentence : “ Hier werden alle mogliche 
Zerfallungen einer Zahl in mehrere Abtheilungen gebracht,” and 
is nothing more nor less than the solution of the problem of 
putting p things in every possible way into n compartments. 
Thus, to take another example, if p were 2 and n were 4, the 
coefficients would be 
2 , 0 , 0 , 0 
0 , 2 , 0 , 0 
0 , 0 , 2 , 0 
0 , 0 , 0 , 2 
1 , 1 , 0 , 0 
1 , 0 , 1 , 0 
1, 0, 0, 1 
0, 1, 1, 0 
0 , 1 , 0 , 1 
0 , 0 , 1 , 1 . 
