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

296 HISTORY OF THE THEORY OF DETERMINANTS

where e use to indicate the summing of the xrsin
obtained by performing simultaneously the cyclical substitutions
JACOBI, C. G. J. (1844, March).
[Sulla condizione di ugualianza di due radici dell' equazion e
cubica, dalla quale dipendono gli assi principali di una
superficie del second' ordine. Giornale Areadico, xcix. pp.
3-11; or Crelle's Joitrn., xxx. pp. 46-50; or (Jesammelte
Werke, i. pp. 271-276.]
By using A, B,.. or be _f 2, ca - g2,...Jacobi first puts
Kummer's sum of squares in a neater form, namely,
15 E(gH - hG)2 + (bF -f B +cF fC -2aF~+2fA )2
+ (bC-cB+cA-aC~aB-bA)2,
and he then gives a lengthy but thorough verification of its
accuracy.
From the fundamental identity
ax2 + by2 + eZ2 + 2f'yz + 2gzx + 2hxy
- L(al X + 31y + YiZ)2 ~ M(a2X + 132'Y + 72 Z)2 + N (a~x + /33't + _y3Z)2
where L, M, N are, as in his paper of 1827, th6 roots whose
reality is to be established,, he obtains at once.
a=La'+Ma ~Na, f = LI31y1+M/3272+N/373y3
b = L0 M2+ It, = Ly/jaj+ My2a2 + Nya,
c= Ly ~My7+Ny~ g = La1~1M 8~a/3
and thence derives
A = MNal + NLa2 + LMa3,F = MNj /~1 + NL/3272 +LM133y3
B =MN /32+NL32 + LM132, GQ= MNy1aj + NLy a2+ ya
C =MN '+NLy 2+LMy, H = MNal/31 + NLa2O32 + LMa3.