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

ALTERNANTS (FRANKE, 1876)

159

the development of such determinants in terms of the t's comes in
for considerable attention in the second section (pp. 216-224).
FRANKE, E. (1876).
[Ueber den Ausdruck, welcher im Fall gleicher Wurzeln an die
Ste]le der Vandermonde'schen alternirenden Function tritt.
Crelle's Journ., lxxxiii. pp. 65-71.]
If the roots of such an equation as
X5-p1X4 +p2X3-p3X2 +p 4x-p - 0
be all unequal, we have already seen how a set of five equations is
obtained for the determination of the p's in terms of the roots
(Hist., ii. pp. 168-169, 175-176, 181-182). When the roots are
not all unequal, differentiation is necessary in order to obtain the
required set: for example, if the roots be
a, a, b, b, b,
the set is
a5 — p +p2a3 - pa2 + pP4a-p = 0)
5a4 - p 4a3 + P2 - 3a 2-p 23 + 2a +p- = 0
b5- p1b4 + p2b3 - p3b2 + p4b - - = 0 -5b4 — P 4b3 + P2 3b2 -p3-2b + p4 = 0
2063 - p * 12b2 + 2'* 6b - p3 ' 2 - 0.
Now Franke's problem is in effect the solution of such a set when
the given equation has
a roots each equal to a,
3 roots each equal to b,
and a -- +... equal to n. He successfully plods his way through
the maze, and formulates at some length the result. The case where
all the roots are alike is used as a test, the common denominator
of the p's then degenerating into
(n-1)! (n-2)!... 2!1! or (n —1)(n —2)2..2"n- 1nas we know it ought to do.