56 
DR. S. CHAPMAN ON THE SOLAR AND LUNAR 
period is a lunar day, and allowing for the slight difference between this and the 
time period by supposing it to have a slowly varying phase. Thus if t in the above 
investigation represents solar time, and t 0 represents lunar time, we may write 
(69) t = £ 0 + i/, t a = t — v, 
where v measures the lunar phase, and increases from 0 to 2- during the interval 
between successive epochs of new moon. The atmospheric oscillation (47) now has 
the time factor sin (r£„ — a) in place of sin (rt — a). Now rt 0 —a = -rt — (a + n). Hence, 
if we replace a by <x. + tv, the above investigation remains unchanged. The result must 
be interpreted in lunar time, however, so that a term pJ'Q.J 1 sin (nt — a) now becomes 
(70) Pm n Qm n sin (nt 0 — a + n— tv). 
Thus the phase of the magnetic variation of the same period (n = r) as the 
atmospheric oscillation remains invariable, while the phases of other components 
increase or decrease by whole multiples of 2 tt during the lunar month. This is what 
is actually observed in the lunar magnetic variation, and the data of Part III. have 
been obtained by allowing for this. It will he noticed, however, that the components 
for which n is negative vary very rapidly in phase (by 2 (n' + r) tt per lunar month, 
n' denoting the numerical value of n). These terms are included in the solar diurnal 
magnetic variation (where their phase is constant), but not in the lunar diurnal 
magnetic variation as here computed. 
§ 23. The Relative Amplitudes of the Magnetic Variations. 
The theoretical results of § 22 will now be applied to the discussion of the relative 
amplitudes of the magnetic variations, leaving the absolute amplitudes and phases 
to be considered later. 
In the case of the lunar diurnal magnetic variation, the nature of the monthly 
changes of phase for the different periodic components indicates that the fundamental 
atmospheric oscillation here concerned is semi-diurnal. The type is not known (§ 20), 
but our assumption of the form Q 2 2 is perhaps not far from the truth. 
According to the theory of § 22, the presence of components of other periods (with 
varying phases) indicates that the electrical conductivity of the atmosphere is not 
uniform and constant throughout the (solar) day. We will first consider what 
evidence is afforded, concerning the nature of the daily variation of conductivity, by 
the relative amplitudes of the harmonics of different periods. 
The simplest law of variation for the latter is given by 
(71) 
1 +a x cos w. 
