288 DR. 8. CHAPMAN ON THE DIURNAL VARIATIONS OF THE 



table is that calculated on the hypothesis />//> = 1+cosw, the second that calculated 

 tV.iin p/pt = l+3co8a>+f cos a w, and the third gives the observed values. The simple 

 f or 1 1 1 of/> gives altogether too small a value for GjC a , while the second expression for 

 p gives values of the right order, at any rate. Perhaps the detailed calculations 

 in Part II. have not been carried to a sufficient degree of approximation, as the 

 expressions for p' do not converge very rapidly. When better data are available, 

 this point must receive consideration. Enough evidence, however, has been brought 

 forward to show that the fourth harmonic component of the lunar variation favours 

 the hypothesis that the conductivity during the night hours is small compared with 

 its value during the daytime,* and that the rate of recombination of ions in the 

 upper atmosphere (assuming this to be the seat of the effect) is rapid, as would 

 naturally be expected. 



The proper discussion of the observations, whether of the lunar or solar magnetic 

 variations, can only be made on the basis of a reliable determination of the numerical 

 coefficients of the various tesseral harmonics in the potential, derived from a number 

 of observatories properly distributed over the globe. The significance of the lower 

 harmonics in the lunar variation makes it desirable to obtain the terms in the 

 potential down to those of the fourth type (Q n 4 ) not only for the lunar variation, 

 but also for the solar variation ; its fourth harmonic shows a sufficient degree of 

 constancy, at most observatories, to entitle it to respect as having definite physical 

 significance. 



PART II. Mathematical Theory. 



15. The problem in hand is to determine the current function of the electric 

 currents induced in a spherical shell of fluid by its quasi-tidal motion across a radial 

 magnetic field of force, the electric conductivity of the fluid at any point being a 

 known function of the angular distance between that point and another (that with 

 the sun at its zenith) which uniformly rotates round the axis of the sphere. The 

 velocity potential t/r of the motion will be expressed as the sum of a number of terms 

 such as 



where Q_ T is a surface harmonic of degree m and type T, and X is the longitude 

 measured towards the east from some standard meridian, at which the local time is t. 

 The colatitude and zenith distance of the sun will be denoted by 6 and o> respectively ; 



* It is not aaserted that any observational evidence has been brought forward in favour of the particular 



numerical constants here chosen for P , but only that the observations indicate the presence of an 



ible term in P depending on cos 2o>, and that this term, if present, may be expected, on general 



physical grounds, to be of such a sign as to diminish the value of P at night as compared with the value 



by day. June 11, 1913. 



