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

AXISYMMETRIC DETERMINANTS (BALTZER, 1857) 147
element in the place (r, s), we have of course
[12] + [13] + [14] ~ [15] = 0,
and consequently also
\/[22] + J[33] + V[44] + \/[55] = 0,
-a result hitherto only obtained independently from geometry.*
BORCHARDT, C. W. (1859, May).
[Ueber eine der Interpolation entsprechende Darstellung der
Elininations-Resultante. Grelle's Jomrn., lvii. pp. 111-121;
or Monatsb. d. Akad. d. Wiss. (Berlin), pp. 376-388; or
Gescaminelte lerke, pp. 133-144; also abstract in Anrnali
di ma....., ii. pp. 262-264.]
Following a suggestion of Rosenhain's, Borchardt seeks to
obtain an expression for the resultant of two equations of the
/tat degree 95(x) = 0, 4r(x) = 0 in terms of the values t which,p(x) and 4r(x) assume when x receives any n +1 values.ao) UV... an.
Baltzer gives (p. 20) Cayley's determinant form for
- (i/a+ J/b+,Q). ( - ~/(i+ /b~ I/C). (~Ia- N/b+ s/c)- (\/a,,/b- /c)-,
placing in front of it what looks like a generalisation, namely
a, b, c,
a, c2 b2
b1 c2 a2
c1 b2 a2
but is not really such. We can easily show that if a1, b,, ce be multiplied and
a., b2, c2 be divided by x, y, z respectively, the determinant is unaltered;
consequently it
aua2 Bb cbcl2 1 1 1. CIC2/ala2 ''6lbz2 V0c2
_ ~2.. 1. Cc2 b~2 _ I~2. I C
b1b2 1 cIc2.aja2 N/~2.ClC2.
c]c2 1 1. 1 b 2b2 aCa2. l1C2 2iJUT 2..aa
HtHe does not mean in terms of these alone, but in terms of these and
aoIal ar