Colloquium publications.

166

THE BOSTON COLLOQUIUM.

which the principal integral is D. might be called an exception, since the continued fraction is then convergent by Pincherle's
definition, but lim N I/D = oo.
A study of the conditions of convergence, so far as I am aware,
has at present been made in only two special cases. Fr. Meyer
[83, a, ~ 7] has made a partial investigation when the coefficients
X\, *..n in equations (6) are negative constants. Pincherle [82]
has examined the case in which the coefficients of the recurrent
relation
fA + (acx + a)J/n+, + b,,^2 =/f+3
have limiting values and finds that the generalized continued fraction is convergent for sufficiently large values of x. Let the limits
of the coefficients be denoted by a, a', and b respectively. To
demonstrate the convergence he avails himself of the notable theorem of PoinearS, already cited in Lecture 4. If, namely, no two
roots of the equation
(15) z3- bz2 - (ax + a')f - 1=0
are of equal modulus, f./f._- will have a limit for n = oo, and this
limit will be one of the roots of the auxiliary equation (15),
usually the root of greatest modulus. From this it follows directly that A/A,1,_ B/B_,, C/ Q_, as quotients of integrals of
the difference equation last given, also Pn/Pn-, QJ/ Qn1, Rn/Rnas integrals of the inverse equation, have each a definite limit. The
existence of limits for Q./Pn and of RI/Pn is then established
for sufficiently great values of x, and the analytic character of
these limits is finally argued. Let them be denoted by U(x) and
V(x). Then X = An + BnU(x) + C V(x) is the principal integral of the difference equation, and has the following distinctive
property: Its expansion in powers of 1 /x begins with the highest
possible power consistent with the degrees of A., Be, C., and
coincides with f for each successive value of n.