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

428 HISTORY OF THE THEORY OF DETERMINANTS

for the determination of g, the main object of his quest. Calling
the determinant here D(t), he takes note of such properties of it as
D( —) = D(;),
D(+ 2v) = D(~),
and of the close resemblance which they bear to those of cos rt.
In this way he reaches without apparent difficulty an important
substitute for his original equation, namely,
Cos 7r = l —V(O),
where V is the determinant obtained from D by multiplying the
central row by -4, the first row on either side of the central row by
4/(42-1), the second row on either side by 4/(82 —1), and so on.
What remains then is the evaluation of V(O). This is accomplished
by removing as a factor from each row the diagonal element of the
row, and then calculating directly the value of the resulting determinant. The question of its convergence as the order tends to
infinity is not forgotten, but no strict investigation is entered on.
It will be observed that this paper of Hill's is very different in
character from those which precede it, and that with it we seem to
pass from matters of languid though legitimate curiosity to one
of serious importance.
ADAMS J. C. (1877).
[On the motion of the moon's node... Monthly Notices R.
Astron. Soc., xxxviii. pp. 43-49.]
Adams opens with an appreciative account of Hill's work, relating
how in his own researches he had followed "in some repsect, a
parallel course, sed longo intervallo." What is interesting for us
to note is that the same infinite determinant had been reached, and
the calculation connected therewith made in 1868 and to a greater
degree of approximation in 1875.
In view of this statement attention may be drawn to a
posthumously published paper of Adams' entitled "Development
of a certain infinite determinant arising in relation to a motion
of the node of the moon's orbit." (Scientific Papers, ii. pp. 85-103.)