The theory of determinants in the historical order of development, by Sir Thomas Muir.

102 HISTORY OF THE THEORY OF DETERMINANTS

SPERLING, I. (1858).
[Teorija oprede'itelej i ej'a vaz'nej'sija priloz'enija.. 1. St.
Petersburg.]
This, dissertation I have failed to see. In English the title is,
The Theory of Determinants and its mnost important applications.
The letters used here in transliterating the Russian title have
German values.
ZEHFUSS, G. (1858).
[Ueber eine gewisse Determinante. Zeitschrift f. Math. u.
Phys., iii. pp. 298-301.]
From two sets of simultaneous linear equations
arx + bry + crz + drW = 0 (r =1, 2, 3,4)
lrX-'+ vry = 0 (r =1, 2)
there arises by multiplication a set of eight equations in
whose determinant, A~ say, must contain as factors the determinants D)4, DA of the original sets. -This observation leads
Zehfuss to consider generally such determinants as
a1l'.2 a1V2 bjx2 bIV2 c1''2 c~iV2 dlAx2 d1V2
a2A''2 a2V2 b21x9 bAv c2'.'2 CAv d2A,'2 d2V2 i.e. D4.2
a3A.'2 a3V2 b31'.2 bAv c3'.'2 CAv c13A2'. d.3v2
a4A,' a4v1 b4,'.'1 b4v1 c4,'.'. c4'.1 d41'.'1 d4v11
a4A'2 a4V2 b4Jx2 b4 V2 c4,''2 cA'' c14,''. d4V2
where the elements of D2 are repeated 4 2 times and each time are
associated with a different element of D4. Multiplying the first
and second columns by DD4/Dal the thi rd and fourth by DD4/Dbj,
the fifth and sixth by DD/c, the seventh and eighth by
DaD4/13dl and performing. two additions we obtain