1898 - 99 .] Dr Muir on a Single Term of a Determinant. 455 
This means that the series of first differences of the numbers in the 
column headed “n” are the same as a certain other series of differ- 
ences in the preceding column. This renders the construction of 
the table still easier. 
(18) ISTo reasonably simple solution of the difference equation 
y ,=v , 
ft, 8 n,8- 
1 
+ V 
71 
1,8 
n-l,8-n 
seems attainable. 
The first form (§ 15) of the equation, however, and especially 
that form when put as a practical rule (§ 16), suggests another mode 
of dealing with the problem. For, when each member of a series 
of numbers is got from another series by adding a fixed number of 
consecutive members of the latter, the process is exactly similar to 
the multiplication of the sum of the members of the series by 
1 + 1 + 1 + . . . Beginning, therefore, with the only number in 
the first column, viz., 1, and multiplying by 1 + 1, then multiply- 
ing the product by 1 + 1 + 1, and so forth, we reproduce the 
columns of the table with ease. Thus — 
1 
1st column 
Multiply by 1 + 1 
1 + 1 
2nd column 
Multiply by 1 + 1 + 1 
1 + 1 
1 + 1 
1 + 1 
1 + 2 + 2 + 1 3rd column 
Multiply by l + l + l + l 
1 + 2 + 2+1 
1 + 2 + 2+1 
1 + 2 + 2+1 
1 + 2 + 2 + 1 
1 + 3 + 5 + 6 + 5 + 3 + 1 4th column. 
(19) On the face of this process there is witness to the truth of 
the theorem of § 7, viz., that 
