MATHEMATICS: HARDY AND LITTLEWOOD 
583 
strain. The usual sex balance has become disturbed. The cause or 
method of this disturbance is not definitely known. But whatever the 
cause, the disturbance influenced not only the individual but also its 
germ plasm and the disturbed balance is evident throughout succeed- 
ing generations. The origin of this disturbance of the sexual balance 
may be referred to as a mutation. While there is little evidence con- 
cerning its cause, there seems abundant evidence concerning its per- 
manent character so far as this strain is concerned. 
The derivation from this sex intergrade strain of several strains 
which produce only normal (so far as may be judged by their morpho- 
logical characters at any rate) females, and, on occasion, normal males, 
is a phenomenon similar to that of the sudden appearance of the sex 
intergrade strain and might with equal propriety be called a return 
mutation. 
I am inclined to believe from evidence from many Cladocera, and 
from other forms reproducing parthenogenetically during most of the 
time and by means of sexual reproduction at irregular and uncertain 
intervals, that environmental factors in all such forms wield the deter- 
mining influence. The evidence at hand in the present case, however, 
is not very conclusive and must be reserved for the larger paper. 
^The following references may be cited: 
Goldschmidt, R., Erblichkeitsstudien an Schmetterlingen. I, Zs. ind. Ahs. — Verer- 
bungslehre, 7, 1-62 (1912); and a preliminary report on further experiments in inheritance 
and determination of sex, these Proceedings, 2, 53-58. 
Riddle, Oscar, Statement run in the Carnegie Inst, Washington Year Book, 12, 322 
(1913); and Sex control and known correlations in pigeons, Amer. Nat., 50, 385-410 (1916). 
SOME PROBLEMS OF DIOPHANTINE APPROXIMATION: 
A REMARKABLE TRIGONOMETRICAL SERIES 
1. The title of this note is perhaps not very appropriate: we retain 
it because the contents of the note form a natural sequel to those of 
three papers which we have published under same title elsewhere,^ 
and in particular those of our second paper in the Acta Mathematica. 
We there discussed in detail the series 
and other similar series associated with the elliptic Theta-functions, 
and used our results to elucidate a variety of difficult points in the theory 
By G. H. Hardy and J. E. Littlewood 
TRINITY COLLEGE, CAMBRIDGE, ENGLAND 
Received by the Academy, August 7, 1916 
(1.1) 
