1910-11.] The Less Common Special Forms of Determinants. 311 
XVIII. — The Less Common Special Forms of Determinants up 
to 1860. By Thomas Muir, LL.D. 
(MS. received June 13, 1910. Read July 4, 1910.) 
There now only remain for consideration those special forms which, 
prior to 1860, had not received any noteworthy attention. These will he 
found to include : (a) permanents, which are touched on by three authors ; 
(/3) determinants with the typical element a rs +b„i, which are referred to 
in four memoirs ; (y) two other forms, which are each dealt with in two 
papers ; and ( S ) nine others, which make their appearance only once. 
(a) PERMANENTS. 
As we have already seen, Cauchy, in his memoir of 1812, widened the 
ordinary meaning of the term “ symmetric function,” and was consequently 
led to call such expressions as 
a 1 b 2 + a 2 b 1 , a A + ajb z + ajb-^ + a A + ajb x + « 3 & 2 , . . . 
“ fonctions symetriques permanentes,” denoting them by S 2 (<xA)> S 3 (<x 1 6 2 ), . . . 
In the same year, as we have also noted, Binet gave the identities 
'lab' = lalb - ab, 
lab' c' = lambic + 21abc-lalbc-lblca-lclab 
which in Cauchy’s notation would have been written 
sw - wy - s w ( a A), 
S W («A C 3 ) = S n (a 1 )S”(6 1 )S w (c 1 ) + 2S"(a 1 & 1 c 1 ) 
but which, in reality, are due to Waring, who, denoting the sum of the 
p th powers of a, /3, y, . . . by s p asserted in his Miscellanea Analytica of the 
year 1762 that 
= - S P+q > 
T, a W = 8 q .s q .s r + 2s p+q+r - ... 
Proofs of Waring’s identities were given by Paoli in his Supplemento agli 
Elementi di Algebra, published in 1804 (see Op., ii. § 28), and by Meier 
Hirsch in his Sammlung von Aufgaben aus der Theorie der algebraischen 
Gleichungen, published in 1809 (see pp. 34-41). 
It is only symmetric functions like S 2 (<x 1 6 2 ), S 3 (cq6 2 c 3 ), S 4 (a 1 6 2 c 3 cZ 4 ), . . . , 
whose every term involves the full number of letters, that at the present 
day are spoken of as permanents. 
