1910-11.] The Less Common Special Forms of Determinants. 327 
Hermite, Ch. (1855, January). 
[Sur la theorie de la transformation des fonctions abeliennes. Comptes 
rendus .... Acad, des Sci. (Paris), xl. pp. 249-254 : or CEuvres, 
t. i. pp. 444-478.] 
The special determinant here considered, as being auxiliary to Hermite’s 
main purpose, is | aj) 2 c z d± | with its elements subject to the conditions 
I <h d 2 I + I b l C 2 I = 0 = I a i d 3 I + I Vs i > ) 
I <qd 4 j + | 6 x c 4 \ = k = \ a 2 d 3 \ + | b 2 c z \ , V 
I a 2 d A I "t I ^2 C 4 I = 0 = | Q> z d^ | + | 5 g C 4 | , 
and the results in regard to it are : — (1) that it is equal to /c 2 ; (2) that the 
row-by-row product of two such determinants is a determinant of the same 
type. No proof is given, but from the way in which Hermite writes the 
conditions, it would appear that the first was obtained by multiplying the 
given determinant columnwise by itself in the form 
d 1 d 2 d z d 4 
-a 1 - a 2 -a z - a 4 . 
A generalisation by Brioschi (1855) has already been dealt with under 
Skew Determinants. 
Zehfuss, G. (1858). 
[Uebungsaufgaben fur Schuler. Archiv d. Math. u. Phys., xxxi. p. 
246 : or Nouv. Annates de Math., xviii. p. 171 : (2) ii. pp. 60-61.] 
The proposition offered for proof by Zehfuss is in modern phraseology 
to the effect that the determinant of the difference of the two square 
matrices 
«1 
. . 
. . 
\ 
b 2 . . 
• • K 
a 2 
a 2 . . 
. . a 2 
h 
b 2 . . 
• ■ K 
a n 
a n . . 
. . a n , 
h 
\ •• 
• • K 
vanishes for all orders higher than the second. The proof given by 
Gustave Harang in the Nouvelles Annates rests on the operations 
col 4 — col 2 , col 2 — col 3 , 
When n = 2 we have 
a \ ~ b i a i~ b 2 
a 2 -b 1 a 2 - b 2 
= («i - « 2 )( & i ~ h) • 
