to Euler. The reader
will find the theory completely treated in Chrystal's _Algebra_, where
will be found the exhibition of a prime number of the form 4p + 1 as the
actual sum of two squares by means of continuants, a result given by H.
J. S. Smith.

The continuant

/ b2, b3, ..., b_{n}\
K ( ) is also equal to the determinant
\a1, a2, a3, ..., a_{n}/

is also equal to the determinant

| a1 b2 0 0 . . . 0 |
| -1 a2 b3 0 . . . 0 |
| 0 -1 a3 b4 . . . 0 |
| 0 0 -1 a4 b5 . . -- |
| |
| u -1 a_{n-1} b_{n} |
| 0 0 -- -- 0 0 -1 a_{n} |,

from which point of view continuants have been treated by W.
Spottiswoode, J. J. Sylvester and T. Muir. Most of the theorems
concerning continued fractions can be thus proved simply from the
properties of determinants (see T. Muir's _Theory of Determinants_,
chap. iii.).

Perhaps the earliest appearance in analysis of a continuant in its
determinant form occurs in Lagrange's investigation of the vibrations of
a stretched string (see Lord Rayleigh, _Theory of Sound_, vol. i. chap.
iv.).

_The Conversion of Series and Products into Continued Fractions._

1. A continued fraction may always be found whose n^{th} convergent
shall be equal to the sum to n terms of a given series or the product to
n factors of a given continued product. In fact, a continued fraction

b1 b2 b_{n}
-- -- ----- can be constructed having for the
a1 + a2 + ... + a_{n} + ...

numerators of its successive convergents any assigned quantities p1, p2,
p3, ..., p_{n}, and for their denominators any assigned quantities q1,
q2, q3, ..., q_{n} ...

The partial fraction b_{n}/a_{n} corresponding to the n^{th} convergent
can be found from the relations

p_n = a_{n}p_{n-1} + b_{n}p_{n-2}, q_n = a_{n}q_{n-1} + b_{n}q_{n-2};

and the first two partial quotients are given by

b1 = p1, a1 = q1, b1a2 = p2, a1a2 + b2 = q2.

If we form then the continued fraction in which p1, p2, p3, ..., p_{n}
are u1, u1 + u2, u1 + u2 + u3, ..., u1 + u2 + ..., u_{n}, and q1, q2,
q3, ..., q_{n} are all unity, we find the series u1 + u2 + ..., u_{n}
equivalent to the continued fraction

u1 u2/u1 u3/u2 u_n/u_{n-1}
-- ------ ------ ----------
1 - u2