1907-8.] Dr Muir on the Theory of Skew Determinants. 303 
XVII. — The Theory of Skew Determinants in the Historical Order 
of Development up to 1865. By Thomas Muir, LL.D. 
(MS. received November 18, 1907. Read December 2, 1907.) 
My last communication in reference to the history of skew determinants 
dealt with the period 1827-1857 ( Proc . Roy. Soc. Edin., xxiii. pp. 181-217). 
The present paper continues the history up to the year 1865, but in addition 
contains an account of two writings belonging to the previous period, 
namely, by Brioschi (1855) and Bellavitis (1857). 
Brioschi (1855, March). 
[Sur l’analogie entre une class.e de determinants d’ordre pair et les 
determinants binaires. Crelles Journal, lii. pp. 133-141. See 
also Annali di sci. mat e fis . , vi. pp. 430-432.] 
After explaining that his purpose is to generalise a result of Hermite’s 
( Gomptes rendus . . . Acad, des Sci., Paris, xl. pp. 249-254) regarding 
determinants of the fourth order, Brioschi sets out by establishing a 
necessary lemma regarding determinants of any even order whatever. It 
is this lemma which is of importance to us in the present connection. 
Taking the determinant 
2( ± fl ll ®22 a ‘2 m,2m) 5 0r A Say, 
he multiplies it by the equivalent determinant 
a i2 
a il 
a u 
CO 
1 
a l, 2m 
~ a l, 2m-l 
^22 
^21 
a 24 
a 23 • • 
. . <X 2 , 2m 
— , 2m- 1 
ni , 2 
~ a 2 m, 1 
&2m , 4 
~ a 2m, 3 • • 
. . CL‘) m t 2 m 
— a 2m t ‘jm— 1 
obtaining the result 
1 21 
l \2 
.... h, 2m 
^32 
^23 
• • • • 1 % 2 m 
• • • • I 3 , 2m 
tm. 
l'2 m, 3 
°2m, 1 
