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

226 HISTORY OF THE THEORY OF DETERMINANTS

where Al(p+l) = (-1)P(2p+1), and the other A's vanish. Another
is given in the second paper, which contains in addition a number
of other recurrents connected with the Theory of Integers.
EUGENIO, V. (1870).
[Considerazioni intorno a taluni determinanti particolari. Giornale
di Mat., vii. pp. 285-290.]
The determinant
x 1.....
2x9 x1 1.....
3x3 x2 xI 1...., or N say,
4x3 x: x2 x
n
we are already familiar with, knowing that it equals
an +3 +y +...
when x, = Ic, x2 = Ya/3, X3 = a3y,... (Hist., ii. p. 211)..
Eugenio considers two other special cases, obtaining
N = (- 1)-l(an + bn), when x,. = (a?'l —b+'l)/(a —b),
N = (-1)n-la, when x. = (a+r-1),.;
and thence making deductions regarding the roots of certain special
equations.
HESS, E. (1872).
[Zur Theorie der Vertauschung der unabhangigen Variabeln.
Zeitschriftf. Math. u. Phys., xvii. pp. 1-12.]
The subject here is essentially the same as that of Bruno's paper
of 1855 (Hist., ii. p. 214), namely, the successive differentiation of a
function of a function. The aims, however, of the writers are
different. Bruno expresses the successive differential-coefficients
of qP {,/ (x)} with respect to x in the form of determinants in which
appear successive differential-coefficients of \r, with respect to x
and successive differential-coefficients of 0 with respect to Vr. Hess,
on the other hand, expresses the last of these sets of differentialcoefficients in terms of the first two sets. For example, denoting
the three sets by
F, F2, F3, * * *.; 1, 2, V3 * *,.; )1, 02, 3, *.