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

MULTILINEANTS (KOTTERITZSCH, 1870)

427

the known solution (Hist., ii. p. 155)
xP = U, ( —1)(U — 2).. * * ( p-aP-1(.-C41)
ap (ap-a)(ap-a2).... (ap-ap-)(ap-ap+.)...
being reached, and there being merely appended the remark that its
usefulness depends on whether the infinite product involved has,
or has not, a finite value for all positive integral values of p.
The second paper is very lengthy and almost equally disappointing,
the new example being that in which ars = 1!(ar+bs), as is the case
in the set of n equations already solved by Binet in 1837 (Hist., ii.
pp. 155-158).
ZEIPEL, V. v. (1871).
We recall the special determinant, Z,.+i say, noted below (p. 458),
which is such that, as r approaches oo, the limit
(2r+l)Zr+1 = 2
HILL, G. W. (1877, 1886).
[On the part of the motion of the lunar perigee which is a function
of the mean motions of the sun and moon. Acta Math., viii.
(1886), pp. 1-36. Original edition published separately at
Cambridge, U.S.A., in 1877.]
In the course of his investigation, Hill arrives at a differential
equation of the second order; and, knowing the integral to be
expressible in the form of an infinite series of multiples of cosines
he substitutes such an expression in the equation, and, as the
condition of satisfaction, obtains an infinite set of linear homogeneous equations connecting the unknown coefficients of the
integral. From this he deduces by elimination the equation
*... (-4)2-0 - - 03 — 0 -0,.
-* 01 (f-2)2- 0o - 0, - 02 -0.
-02 -01 2_ -01 -02... = 0
- 03 - 0 - 1 ( +2)-0 -0...
* * ~ -04 -03 -02 0 -01 (C+4)2 -0o...........................