1902 - 3 .] Dr Muir on Axisymmetric Determinants. 
563 
and that the coefficient of - x° in the cubic is the square of A, 
the coefficient of x 1 the sum of the squares of what came afterwards 
to he called the “ primary minors ” of A, and the coefficient of x 2 
the sum of the squares of the secondary minors. 
Jacobi (1832). 
[De transformatione et determinatione integralium duplicium 
commentatio tertia. Crelle's Journ ., x. pp. 101-128.] 
This last paper of the three dealing with the transformation of 
integrals contains less regarding our present subject than either 
of the others. The only thing worth noting is the curious cubic 
equation 
x* 
+ x 
{ abc - ad 2 - be 2 - ef 2 + 2 def } 
( a (be - d 2 ) + b'(ea - e 2 ) + c\ab - f 2 ) 
t + 2 d\ef - ad) + 2 e(fd - be) + 2f(de - ef ) 
j a(b’e - d' 2 ) + b(ca - e 2 ) + c(ab’ - f 2 ) 
1 + 2 d(ef - ad’) + 2e(fd' - b’e) + 2 f(d'e - ef) 
{ a'b'c - ad’ 2 - b’e 2 - ef 2 + 2d’ef} 
= 0 , 
where the first and last coefficients are in modern notation 
[ a f e 
a f e 
f b d 
f b' d! 
e d e 
, 
e d' e 
the second coefficient from the beginning is 
a 
f 
e 
a f 
e 
a f 
e 
f 
b 
d + 
f v 
d' 
+ 
f b 
d 
e 
d 
e 1 
e d 
e 
e d’ 
c 
A a + B V + Ce + 2Dcf + 2Ee' + 2F/ ; 
and the second from the end 
a f e 
a f e 
a f e 
f b’ d’ 
+ 
f b d 
+ 
f b’ d! 
e d’ c 
e d! e 
e d e 
A a B b + Cc + 2D d + 2E e + 2Fy , 
Att + Fy + E e ) 
+ F f + BA + D ’d > 
+ E'e + D’d + C'c 
