be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. {368} positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be A/B, where A and B are integers. First, whatever this limit may be, it lies between -1 and +1. This is obviously the case with any fraction p/(q + [omega]), where [omega] is between +-1: for, p/q, being < 1, and p and q integer, cannot be brought up to 1, by the value of [omega]. Hence, if we take any of the fractions a/b, a/b+ c/d, a/b+ c/d+ e/f, etc. say a/b+ c/d+ e/f+ g/h we have, g/h being between +-1, so is e/f+ g/h, so therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h. Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being integers. Let P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc. P, Q, R, etc. being integer or fractional, as may be. It is easily shown that all must be integer: for {369} A/B = a/b+ P/A, or, P = aB - bA P/A = c/d+ Q/P, or, Q = cA - dP Q/P = e/f+ R/Q, or, R = eP - fQ etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. Now there can be no such _unlimited_ succession of _descending_ integers: consequently, it is impossible that a/b+ c/d+, etc. can have a commensurable limit. It easily follows that the continued fraction is incommensurable if a/b, c/d, etc., being at first greater than unity, become and continue less than unity after some one point. Say that i/k, l/m,... are all less than unity. Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be [kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f + [lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b + [nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... itself; which is therefore incommensurable. Let [phi]z represent a a^2 a^3 1 + - + ------- + -------------- + .... z 2z(z+1) 2.3.z(z+1)(z+2) {370} Let z be positive: this series is convergent for all values of a, and approaches without limit to unity as z increases without limit. Change z into z + 1, and form [phi]z - [phi](z+1): the following equation will result-- a [phi]z-[phi](z+1) = ------([phi](z+2)) z(z+1) a [phi](z+1) a [phi](z+1) a [phi](z+2) or a = - ---------- . z + - ---------- . --- ---------- z [phi]z z [phi]z z+1 [phi](z+1) a = [psi]z(z+[psi](z+1))