148 
Proceedings of Royal Society of Edinburgh. [sess. 
On this are performed the operations 
colj — col 2 , col 2 — col 3 , col 3 — col 4 , 
the result being a determinant of the next lower order 
3a -r -2 r - 2a - 1 2 a 
- 2-2 a 2a - r r - 3a - 2 
1 - a a -r +2 
Finally, after changing the signs of all the elements here, Jthe 
operations 
row 4 + row 2 + row 3 + . . . , row r 2 + row 3 + . . . , row 3 + . . . , ... 
are performed, the result 
r -a a 
2(a-l) r-a-1 2a 
a - 1 r — a - 2 
being a determinant exactly similar in form to the original hut 
with r — a instead of r. This, therefore, in turn may be trans- 
formed into 
(r + a- 2) 
r - 2a 
a - 1 
a 
r - 2a — 1 
and so on. 
The value thus obtained for the above-written determinant of 
the (n+ l) th order is 
(r + na - n)(r + na-n- < 2a + l)(r + na — n - 4a 4- 2) . . . (r - na) 
each factor being less than the preceding by 2a- 1, and the whole 
a function of a(a - 1). 
The special case is noted where a = J, and where therefore all the 
n + 1 resulting factors are alike. This Painvin writes in the form 
r 
n 
2 
1 
2 
r - 1 
2 
2 
n - 2 
~ 2 ~ 
3 
2 
r - 3 
n\ 
-2 ) 
. . . r -n+1 
n 
2 
1 
2 
r - n 
