EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN. 287 



adapted, in J; L':;. |.> n,\ t -r t In- case ..(' t In- luiii solar ina^m-t ic \ ariat inn*. It istlirn- 

 shown that the equivalent expression to (I>) is in thin case (apart from a constant 

 factor) 



This expression, it should be noticed, consists of series of harmonic components of 

 one, two, three, and more periods in the lunar day, with phase angles which depend 

 on the age of the moon. In the second series the phase angles increase by 2 (<r+2) * 

 per lunation ; this phase change is very rapid, even for the diurnal term, and with the 

 lunation divided up into not more than eight parts, hardly comes within the range of 

 observation, even if the coefficients q* were of the same order of magnitude as the 

 p coefficients. The theoretical values of q m ' are, however, much less than those of 

 the important members of the p n ' set of coefficients, and therefore this part of the 

 magnetic potential can be neglected. The other part consists of terms of period 2iw, 

 whose phase angles increase by 2 (a- 2) IT per lunation ; thus the phase of the first 

 harmonic decreases by 2-ir each lunation, that of the second component remains 

 constant, while the third, fourth, and higher components increase by amounts 2ir, 4ir, 

 6-r, and so on. This, however, is exactly the law of phase change which is indicated 

 by the formula (B), whicli was determined empirically from the observations. 



At new moon, when v = 0, the formula indicates that all the harmonic components 

 should have the same phase angle, or differ by 180 degrees exactly (since the 

 coefficients may be of different sign). The data obtained in this paper show a very 

 satisfactory agreement with this conclusion, when the extreme smallness of the whole 

 phenomenon is considered. 



13. The amplitudes must next be considered. The actual calculations necessary 

 for the comparison of theory and observation are given in 25, and only the results 

 obtained will be cited here. It appears that as regards the relative magnitudes of 

 the first three components in the lunar variation, there is tolerably good agreement 

 with the results derived either from SCHUSTER'S simple theory p/pi = l + cosw, or 

 from the more general theory of Part II. of this paper. The numerical constants 

 (p/p<>= 1 + 3 cos o> + Jco8*w) might be altered to fit the observations better, but it 

 seems hardly worth while to do this till better observational material,, is available. 

 The given constants were chosen to represent a function which should have a large 

 maximum at midday, and should be small and nearly constant during the night 

 hours. 



J 4. The deciding factor between the two expressions for pfp^ is found to be the 

 amplitude of the fourth harmonic component. Three tables are given in 25 to 

 illustrate this. They give the ratio of the amplitudes of the four harmonic compo- 

 nrtits to that of the second component, for the three elements X, Y, Z. The first 



