"If in the infinite continued fraction of the second class a_n [>=] b_n + 1 for all values of n, it converges to a finite limit not greater than unity." 3. _The Incommensurability of Infinite Continued Fractions._--There is no general test for the incommensurability of the general infinite continued fraction. Two cases have been given by Legendre as follows:-- If a2, a3, ..., a_n, b2, b3, ...,b_n are all positive integers, then b2 b3 b_{n} I. The infinite continued fraction -- -- ----- converges a2 + a3 + ... + a_{n} + ... to an incommensurable limit if after some finite value of n the condition a_{n} [not <] b_{n} is always satisfied. b2 b3 b_{n} II. The infinite continued fraction -- -- ----- a2 - a3 - ... - a_{n} - ... converges to an incommensurable limit if after some finite value of n the condition a_{n} [>=] b_{n} + 1 is always satisfied, where the sign > need not always occur but must occur _infinitely often_. _Continuants._ The functions p_{n} and q_{n}, regarded as functions of a1, ..., a_{n}, b2, ..., b_{n} determined by 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}, with the conditions p1 = a1, p0 = 1; q2 = a2, q1 = 1, q0 = 0, have been studied under the name of _continuants_. The notation adopted is / b2,...,b_{n}\ p_{n} = K ( ), \a1, a2,...,a_{n}/ and it is evident that we have / b3,...,b_{n}\ q_{n} = K ( ). \a2, a3,...,a_{n}/ The theory of continuants is due in the first place 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.).