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

220 HISTORY OF THE THEORY OF DETERMINANTS

and treating the first four columns of the numerator in the manner
set forth in the earlier part of his paper, he obtains
1... 1
l 1.. x
2 1 x1 = x- ax + (...X
53 2 8I 1 x
54 3 6s2 S1 X4
whence are got, by equating coefficients, the determinant expressions for la, la/,.... in terms of the alephs. The two other
modes of expressing the same relations follow in the way one would
expect.
HANKEL, H. (1865).
[Darstellung symmetrischer Functionen durch die Potenzsummen.
Crelle's Journ., lxvii. pp. 90-94.]
Hankel's theorem may be put in the form
X...) (A?) = (
x~ n n )

wher

e An== - Sa 1.....
8la+r3 82 2
Sa+./+7 83+y 8Iy t...
Sa+P+.... +.y+. -. +v S8+.. +v...
and the items comprised in Y'(An) are obtained by permutation of
a, f3, y,..., v, and those comprised in l"(An) by permutation
of all except v. For example,
yXIxy-i = 21 z ( 1 ' 1
Z^1^xsi ~ / S,, s~ 2 I

sv

s+3tt+y +y
8 8++, P/3+y $

i

= (S y+2S +y Sa+py-25p+yca)
-r SaS3Sy +2 Sa +, ++ y-s +8y - 2sp +,ys_
= FSS.Sy 8t + +y - Sa+A y - a +38, a - 2 S+y5,