The amplitude r2 and the hour of maximum ϕ2 of the second harmonic of the neutron intensity at Deep River and Chacaltaya are studied for the period 1956--1968. It is shown that r2 can be significant (exceeding 0.23%) at the both of these locations, not necessarily simultaneously. Also, the space direction of the second harmonic is neither field aligned nor perpendicular to the interplanetary magnetic field on many occasions. Neither of these facts agrees with the theoretical models presently in vogue. The possibility that r2 is a parasite of r1, owing to the possible limited sinusoidal nature of the space anisotropy, is examined, but the explanation is found to be only partly valid. On individual days, sharp decreases, particularly in the morning hours, are often observed, indicating sinks (rather than sources) as possible causes of the semidiurnal as well as the diurnal components. Highly diurnal patterns are sometimes observed at the equator as well as at high latitudes, indicating that the semidiurnal component is by no means a permanent feature in any region. As the daily values of r2 can occasionally be as high as 0.5% and very frequently are 0.2% or more (all are statistically significant), average yearly values of the order of 0.05% or less used for theoretical explanations seem to be of dubious value. There are indications that the energy spectrum of the daily variation patterns may not remain constant throughout the day.