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

RECURRENTS (WOLSTENHOLME, 1867)

223

WOLSTENHOLME, J. (1867).
[A BOOK OF MATHEMATICAL PROBLEMS,.344 pp. London
and Cambridge.]
Problem 922 concerns the determinant whose first row is
1-X1, X1(1-X2), X1X2(X3),.... X2... X
and whose (-+I)t" row is in every case got from the rtli by
differentiating with respect to x,r The value of the determinant is readily seen to be 1.
SARDI, C. (1867).
[Nota sni numeri primi. Giornale di Mat., v. pp. 371-376.]
Defining k,,,2, by the identical equation
(x+l)(x+2)... (xI-n) X X"_'+kvJX?'~~,~X~- a
Sardi establishes the relations
kit, I - (n+1)2 = 0,
2 k, 2_()2k,,(n+)3= 0,
and thence obtains
(n+1)2 -1
(n+ 1) (n)2 -2.
I (n+l),4 (n), (n
(n + i) (n)31 (n -1).,.(n-p +3)2 -(p-i)
(n-Ili)p+, (n)p (n -i)p... (n-p-p3)2 (n-p+2)2
from which his results regarding primes are dednced.
It must be noted, however, that almost a century earlier the said
set of relations was ready to hand for the construction of a recurrent,
having been reached by Lagrange in 1771.*
* LAGRANGE, J. L. Demonstration d'un th'or'me nouveau coneernant les
nombres premiers. Nouv. 11M1m.... Aced. des Sci. (Berlin), An. 1771, p. 125;
or (Euvres completes, iii. pp. 422-438.