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

CONTINUANTS.(SMITH, 1854)42

425

SMITH, II. J. [S.] (18-54, May).
[De compositione numerorum primorum formae 4X +1 ex duobus
quadratis. Crelle's fourn., 1. pp. 91-92; or Ciollected Math.
Papers, i. pp. 33-34.1
Pointing out, as Sylvester had already done, that the writing
of Euler's algorithm (q,, q2 qn) as a determinant leads easily,to the properties
(q1, q72,.. qi) = (qi, qi-1.. I q0)q+1 I
Smith therefrom deduces that- f or centro-symmetric series of
elements
(gj, q2, I q I,7 I q2, qI) = I 1 q2,.,cj)2-t-(q1, q2,. i, 2
and
(ql I q2,. L~.. )i1 q i1, q1) = (qI I q2) qi-1){~q1,. qj)-I-(q1,. i,
noting in regard to the, former that the two numbers squared on
the right are mutually prime. He then makes appliceation to
the theorem referred to in the title of his paper.
SYLVESTER,- J. J. (1`854, August).
[The'ore'me sur les determinants de M. Sylvester. Nouv. A nnales
de, Math., xiii. p. 305; or Gollected M1ath. Papers., ii. p. 28.]
This communication in its entirety is as follows:
"Soieut les de'terminiants
X, A 1 Ak 1 0 A 1 0 0
1 A, 2 A 2 3 A 2 0
0 1 AI 0 2 A 3
0 0 1 A,
I10 00
4AX20 0
0 3 3 0
0 02 A4
0 00 1 X