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

CONTINUANTS (HEINE, 1858)

435

a11x12 + 2a12xx2 + 2a13X13 +...
+ a22X22 + 2a23x2x3 +.
+ a33X32 +..
be transformed by an orthogonal transformation into
Allf2 + A22 22 + A33 32 +. *
the coefficients of the latter expression are the roots of the
equation
all- A 12 al3...
a12 "22-A a3 = A
a13 a23 a33 A
Now Heine's peculiar quadric was
Co2302- 2KCOClX0X1
+ (el2 + c22) x2 - 2,cc2c3xlx2
+ (c32 + C42)22 -
+ (c,-l +c )x0,5
where in every case the coefficient of the product of two x's
vanishes if their suffixes differ by more than 1, and where
Co2 = 2(,) (n + 1),
2 = 1 (-1)( + 2),. I
c,2 = n(~-r)(n++r 1), (r > O)
C2n-1- 2!

c2 - b2
and K = n+ -
He was thus naturally led to the equation in z
Z-Co2 KCC1..
KC01 Z -C12 - C2 KCC.
C2C3 -C32-C42........ KC
Z

= 0,
2 o-1
L ~ ^2<r

2o - 20C
C2 2o - 1