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

190 HISTOIRY OF THE THEORY OF DETERMINANTS

ZEHFUSS, G. (1859).
[Ueber die Determinante Q6 = 1 ~ (a0 + bo)P(cl + b1)P... (a,, + bn)p.
Zeitschrift f. Math. m. Phyjs., iv. pp. 233-2.36,1
Zehfuss having drawn attention to the known results when
p = - I and p = - 2, and having asserted that there are no similar
simple results when p is less than -2, confines himself to the
cases where p is a positive integer. He first shows in the usual
way that Qp is divisible by the difference-product of a., al..., an
and by the difference-product of bo, bl,..., b%. Then he points
out that Qp is of the pth degree in a0I while the differenceproduct of a0, a,,..., a. is of the nlth degree, and thence draws
the conclusion
ifp<n Q= 0,
giving the example deduced above from Cayley,
~,-i (ao + bo)2(al ~ b,)2(a2 b9)2(a3 + b3)2 = 0.
The case where p = n he deals with differently, namely, as
might have been suggested by Cayley's paper of the preceding
year. By the multiplication-theorem he derives
2
1 'a 2a.... a0 b b
n, a, b~n Vn
Qnn~jl nla1 u~a1.... a1 bi z bil'.... 1
Qn i n~a2 n2a2.... 2~ b2 b21..*
1 a a. a b. 1
nia,, zc a' b b12
where n, n(n- 1)(n-2).... (n-r+1) 1.2.3 r...; and from
this there is readily obtained
Qn= (-1 2 1)' 3. (a0a1... an). - (bobt... bn)
the illustrating example being again that which we have
deduced above.