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

CONTINUANTS (MUIR, 1877)

419

MUIR, T. (1877).
[Note on determinant expressions for the sum of a harmonical
progression. Proceed. R. Soc. Edinburgh, ix. pp. 361-363.]
The progression is transformed by Euler's process into a continuedfraction, and thence into the quotient of one continuant by another.
The second of these, in view of previously obtained factorizable
continuants (Sylvester 1854, Painvin 1858) is worth noting, namely.
a a2...
1 2a+b (a+b)2.
1 2a+3b (a+ 2b)2.... = a(a+b)(a+2b)(a+ 3b)
1 2a + 5b....
MARTIN, A. (1877).
[Question 5484. Educ. Times, xxx. p. 197; or Math. from Educ.
Times, xxix. pp. 90-91.]
The theorem here involved, though stated differently, is that the
product of any two of the expressions
1 + K(b,a,a,b), 1 + K(c,b,a,a,b,c), 1 + K(, c,b,a,a,b,c,1)
diminished by the sum of the same two is a square. It is not pointed
out, however, that the three squares in question are the squares of
K(b,a, a,b,c), K(b,, a,ab,c, 1), K(c,b,a,a,b,c,1).
S+A+ sc, - b
2 -+ A + sc2 b3 3 ~
84 5
s,
83 SZ64
and specializing further by putting each of the s's equal to s we come back to the
theorem of the paper above reviewed.