224 Proceedings of the Royal Society of Edinburgh. [Sess. 
and the multipliers are 
C 2 A -1 + 
k(2n - 4& - 3) 
1) 
C2*+l 
(2 n - 2 k ■ 
k + h-1 
k-l .\h ' (2w - 2k + l){2n — 2A? — 1 ) . . . (2% - 2k + 1 - 2 . h^l) 
(2 n - 4& + 3)(2?i - 4& + 1) . . . (2% - 4& + 3 - 2 . A - 1 'jp . 
V = r ; C2fc+2^-l + 
C 2 & + 
&(22i-4&+l)^ 
( 2 tt - 2 £- l )° 2A:+2 + ‘ * ‘ 
| |^ + 7t ~ 1 (2?t-4A:+l)(2?i-4fe-l). . .(2n-4&+l-2.~fe^l ) 0; ^ + 
lAzl* I* ’ (2n-2fc-l)(2n-2&-3). .. (2»-2&-l -2.1^1) 2 * +21 
(A-l)(2w-4A + 5) 1 
+ (-> 
(2n-2* + l) 
| fc-l 
-R 2&-2 + • • • 
(2?i-4& + 5 + 4 . h 
1)(2 n - A.k + 3)(2w- - 4fc + 5) , 
b»+i 
| k-h-l , \h 
k(2n - 4^ + 3)t, 
■KsSfc-r 
(2 n 
{2u — 4k -f- 3 + 2 . li — 2)p . 
: -&2k-2h + 
(2 n - 2& + 1) 
„ \k 
+ (~) h - 
{2n-2k + l){2n-2k + Z). . . (2?i - 2&+ 1 +2 . h- 1) 
4fc+3+4. fr-l)(2tt-4& + l)(27i-4fr + 3). . . (2ro-4fc + l + 2. ft-2 ) R ^ + 
| ^ (2w-2fc + l)(2w-2& + 3). . . (2?i-2fc + l + 2. fc-1) 
where C* and R, represent the ith column and jth row respectively. 
3. The work of finding these line-multipliers will not be given here, 
though a few words as to the method used may be of interest. A series of 
consecutive non-zero elements along and parallel to the principal diagonal, 
beginning with that in the 2rth row and 2rth column, were written down, 
and the various multipliers for these general rows and columns determined 
under the conditions that when all the operations were completed the result- 
ing determinant was such that all the elements, say below the principal 
diagonal, were zero, except those in the odd places of the line immediately 
below the principal diagonal, in which case the determinant obviously breaks 
up into quadratic factors, each of which is the difference of two squares. 
4. As an illustration take the determinant of order eight 
a 0 ! 
ft + 6 6(0 + 1 
- o 
2(0 + 5) 
-3 
3(0 + 4 ) 
- 1 
5(0 + 2) 
4(0 + 3) 
1 
4(0 + 3) 
3(0 + 4) 
1 " -1 
5(0 + 2) 2(0 + 5) 
a -3 
3 
6(0 + 1 ) 
0 + 6 
-5 
0 
