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

ALTERNANTS (MURPHY, 1832)

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which he neatly and easily solves, giving the value of x,, and
thus in effect evaluating
1 1 1.... 1 1.... 1 1.... 1
1 2 22.... 2-1 2m+1... 2. 2 22....2n
()m 1 3 32....3- 3ln+l.....3 3 32... 3n1.1..2.2...m- 1 m+l...........7
His connectionwith our subject is thus seen to be similar to Prony's.
It should be carefully noted, however, in passing, that Prony's
set of equations is not the same as Murphy's, the determinant of
the one being conjugate to that of the other.* When the use of
determinants is debarred or avoided, this difference is far from
unimportant,-a fact which might readily be surmised from the
present instance, since Murphy's mode of procedure, though
strikingly effective upon his own set, is quite inapplicable
to Prony's. It should also be observed that the solution
of Murphy's set is not essentially different from the solution of
the familiar interpolation-problem to determine a, a,.....,
so that a1 + a2S + a,32-+.... + a2- n 1 or y?may have the values
Y, Y2,...., n Y when x has the valuces xl, x2,..., x 1 respectively,-a problem which had been solved in one way by
Newton (1687), in another way by Lagrange (1795), and in a
third way to a certain extent by Cauchy (1812).t
* The two sets of equations are
1 2 ralxl + ar2 +... + ax,,=.} (I)
and A?-r=
nd alx + ax22 +... -+ a,, - = t (J)
The former is substantially the set of the interpolation-problem which goes back
to Newton, and which may therefore for distinction's sake be associated with
his name: the latter being first found solved by Lagrange (Recherches sur les
suites recurrentes... Mlm. de l'ccad. de Berlin, 1775, pp. 183-272; 1792,
pp. 247-257: or (Euvres completes, iv. pp. 149-251; v. pp. 625-641) may be
called Lagrange's set, provided we remember that he also gave a solution of
the other. The first to deal with both of them in more or less general form
by means of determinants was Cauchy (1812)-see also his Re.smens Analytiques,
p. 19,... 4~ Turin, 1834-but in saying so a mental reservation must be
made in view of Cramer's mode (1750) of continuing Newton's work.
f NEWTON, Principia, lib. iii. lemma v.: also Arithmetica Universalis,
probl. lxi. LAGRANGE, Journ. de l'ec. polyt., ii. cah. 8, 9, pp. 276, 277; or
(Euvres completes, vii. pp. 285, 286. CAuCHY, Journ. de 'c'e. polyt., x.
cah. 17, pp. 73-74; or (Euvres completes, 2e ser. i. pp. 133-134.