EARTH'S MACM 1IS.M l'l:oi>(JCEP BY THE MOON AND SUN. ::<'.. 



The following are the values of the corresponding tesseral harmonics: 



Qi'ssind, Q, 1 = 3 sin 6 cos 0, Q, 1 = i sin 0(5 cos' 6-1), 



Q, 1 = i sin 0(70 cos 3 6- 15 cos 0), Q* 1 = g sin 0(63 cos 4 0-84 cos* 0+3), 



Q a 3 = 3 sin a 0, Q, s = 15 sin*0 cos 0, Q 4 S = \* sin'0(l4 cos 8 0-1), 



Q s = J-lp sin* (3 cos 3 0-2 cos 0), 



Q 3 S = 15 sin" 0, Q< = 105 sin 8 cos 0, 06' = H' A n* (9 cos* 0-2), 



Q/ = 105 sin 4 0, Q 6 = 945 sin 4 cos 0. 



Since all the stations for which we have observational data, in Part III., are 

 tropical, we shall consider the values of X, Y, and Z for such stations only. Hence, 

 in our expression for V, the magnetic potential (which we must now use instead of 

 the current function), all terms containing cos s may be neglected, and will be 

 omitted. Thus we get 



CT /. - ,, sin cos ,, . . sin0\ /., , \ 



V = [H COBS. - + T gJ 8>"<*C08 S )c08(\' + <-a), 



\ r r I 



,//,., ., - j-i Txsin a 0cos0 l7] . sin* 01 / , ^ 

 + {( + i I 3 cos" S- 1) - - - + -fts sm S . -pp-j cos (2X'-a), 



. ,sin 3 0\ /./ \ 



sindcosd )cos(3\ v a), 



2 . sin 4 CO80 / .. , \ 

 cos o - ~j -- OP 6 (4X 2c a). 



In the above expression, the terms depending on sin S represent the main seasonal 

 effect. Since 



Y v 



aX = -r , aY = . 



' 



-r , . n . , -- =-, 



df) ' BmOdX dr 



it is evident that when cos0 is put equal to zero after the differentiation, only the 

 terms in V which do not contain cos will contribute any result to Y and Z. But 

 the above equation shows also that these terms always contain sin S, so that at 

 equatorial stations Y and Z change sign in passing from summer to winter. 

 Tables XL and XIII. corroborate this sufficiently well, especially when it is 

 remembered that the stations are not quite equatorial, and that the obliquity of the 

 magnetic axis also produces a disturbing effect. A further interesting consequence 

 of the above equations is to indicate that at the equator the terms in X which 

 depend on sin <?, i.e., the seasonal terms in the horizontal force variation, vanish. 

 This agrees with the known fact that at tropical stations the X variation hardly 

 'lianas throughout the year. Table XII. illustrates this, especially for the most 

 nearly equatorial observatory, Batavia (6 S.). 



Km comparison with observation we shall write down the values of the ratios 

 of the amplitudes of the first, third, ami fourth harmonic components to that of the 



vol.. i . AMI. A. 2 R 



