1888-89.] Dr T. Muir on the Theory of Determinants. 
225 
There has been no previous instance of an identity perfectly similar 
to this; the nearest approach to such being, as the numbering shows, a 
result obtained by Binet in 1811. The exact character of the affin- 
ity between the two, and the general theorem which both foreshadow, 
will be most readily brought into evidence by a little additional trans- 
formation. Taking first the right-hand side of the identity, we ob- 
serve that the three determinants have only twelve elements among 
them, being obtainable in fact from a single array of four rows and 
three columns. Their sum may consequently be put in the form 
1 (Va) (x m X b ) (Vc) I 
1 (x n x a ) (x n x b ) (x n x c ) 
1 (x p x a ) (x p x b ) (x p x c ) 
0 1 1 1 
Secondly, we observe that the first factors on the left-hand side are 
similarly obtainable from 
m 
1 
X» Vn Z n 1 
Xp Vp Z p 1 ; 
and the second factors from 
Xa Va Z a 1 
x b Vb % 1 
a? c Vc z c i ; 
and as the so-called product of these arrays is equal to the said left- 
hand member diminished by 
x m 
y m 
z m 
x a 
y a 
z a 
X n 
y n 
z n 
x b 
yb 
Zb 
x p 
y P 
Z p 
x c 
y c 
Z c 
Mainardi’s theorem may be put in the much altered form- 
(x m x a ) (x n x a ) (x p x a ) 
M (x n x b ) (x p x b ) 
(x m x c ) (x n x c ) (x p x c ) 
1 1 1 
X m V m Z m 
X n y n 
Xp y.p z p 
X a y a Z a 
Mb yb %b 
x c y c *c 
x m 
y m 
z m 
y a 
Z a 
x n 
y n 
Zfi 
• 
X b 
y b 
z b 
Xp 
yp 
Zp 
x e 
y c 
Z c 
VOL. XVI. 
10/7/89 
