1910-11]. The Less Common Special Forms of Determinants. 325 
the m’s are all less than 1, and a n+l — oo . The simplest example is 
-» dx 
{x) 
(v* r 
i *•<*) i 
j 1 e . a _xdx_ j 
I <£ 0 (z) I </). 
^«0 ° •/«! 1 
xdx 
¥) 
= T(1 - m 0 ) . r(l - m x ) . (a x - a 0 ) l ~ m o _m i . e - a o - «i 
In establishing the result, use is made of the fact that the determinant is 
expressible also as a multiple integral : for example, the two-line deter- 
minant just written is equal to 
e~ x ~ x i(x 1 — x)dxdx x 
( a - x 0 ) m o(a x - x ) m i(x 1 - a 0 ) m o(aq - a x ) m i 
Bazin, [H.] (1854 July). 
[Demonstration d’un theorem e sur les determinants. Journ. ( de Lion - 
mZZe) de Math., xix. pp. 209-214.] 
The theorem in question is to the effect that if there be two n-hy-m arrays 
R, R 7 with integral elements, and such that the ratio of any r^-line minor 
of R to the corresponding minor of R 7 is constant and integral, and if the 
ii-line minors of R have 1 for their highest common factor, then it is 
possible to find a determinant S of the v} h order with integral elements so 
that the product of S by any 7^-line minor of R 7 shall equal the correspond- 
ing minor of R. For example, it being given that 
k 
&i b 2 6 3 
*1 
X 2 
X 3 ! 
C 1 C 2 C 3 
Vi 
y 2 
Vz 
where all the letters denote integers, and that the highest common factor 
of | \c 2 1, | \c 3 |, | b 2 c 3 | is 1, four integers a, f3, y, S can be found such that 
j a P 
b x b 2 b 3 
x \ X 2 X 3 
y B 
C l C 2 c z 
Vi y 2 Vz 
Brioschi, F. (1855). 
[Additions a l’article No. 15, page 239 de ce tome. Crelle’s Journ., L. 
pp. 318-321 : or Opere mat., v. pp. 271-276.] 
The determinant here (pp. 320-321) dealt with is, for shortness’ sake, taken 
to be of the 4th order, namely, | m x S 2 b s c 4 1, in which <5 1? S 2 , S 3 , S 4 stand for 
az 1 + ex i +fz 3 + gx 4 , 
ex x + bx 2 + hx z + kx 4 , 
fx 1 4- hx 2 + cx 3 + Ix 4 , 
gx x + kx 2 + lx z + dx 4 , 
