360 
MATHEMATICS: D. N. LEHMER 
The possibility of some form of dissociation is suggested by the harmonic 
relationship between the displacements of the components of the hydrogen 
lines, and the early appearance of the nebular lines in the spectra of these 
stars adds interest to considerations of this nature. 
1 Adams, W. S., and Kohlschiitter, A., Mt. Wilson Contr, No. 62, Astroph. J., Chicago, 
III., 36, 1912, (293-321). 
2 Campbell, W. W., and Wright, W. H., Lick Obs. Bui. No. 8, Berkeley, Cat., 1901. 
3 Schemer, J., Astronomical Spectroscopy (Frost), Boston, Mass., 1894, p. 290. 
4 Michelson, W., Astroph. J., Chicago, III., 13, 1901, (192-198). 
5 Paddock, G. F., Pub. Astr. Soc. Pac.,San Francisco, Cal., 30, 1918, (244-249), p. 249. 
ON JACOBVS EXTENSION OF THE CONTINUE!) FRACTION 
ALGORITHM 
By D. N. Lehmer 
Department of Mathematics, University of California 
Communicated by E. H. Moore, July 23, 1918 
It has been known since Lagrange 1 that the regular continued fraction which 
represents a quadratic surd becomes periodic after a finite number of non- 
periodic partial quotients, and conversely, a regular continued fraction which 
becomes periodic after a finite number of non-periodic partial quotients is 
one root of a quadratic equation with rational coefficients. It is useless, 
therefore, to look for periodicity in regular continued fractions which repre- 
sent cubic and higher irrationalities. To meet this difficulty Jacobi 2 under- 
took to extend the continued fraction algorithm as follows: 
In the case of the ordinary continued fraction we are concerned with two 
series of numbers, A n , B n , (the numerators and denominators of the successive 
convergents) which are given by the recursion formulae 
A n = q n A n -i + A n -2, B n = qnBn-l + B n -2, 
with the initial values A 0 = 0, A-i = 1, B 0 = 1, = 0. Jacobi con- 
siders three series of numbers, A n , B n , C n , which are given by the recursion 
formulae 
An = pnA n - 1 + q n A n - 2 + A n _ 3, 
B n = pnB n - 1 + Q_nB n - 2 + B n - 3, 
C n = pnCn - 1 + QnC n _ 2 + C n - 3, 
with the initial values: 
(A _ 2 , A _x, Ao) = (1, 0, 0), (B- 2 ,B- h B) = (0, 1, 0), 
(C-2, C-x, Co) = (0, 0, 1), 
Jacobi then chose the coefficients of the expansion, p n , q n , so that A n : B n : C n 
should approximate 1 : 6: a + bd + cd 2 where 6 is the real root of a cubic equa- 
