EARTH'S MAGNETISM PRODUCED BY THE MOON AND SUN. 



297 



li). Consider now the terms .-irising ('nun a term A J .. T Q m . T ' sin (r'X'ct) in >k. The 

 only term of corresponding type involving d a , on the right-hand side of equation (4), 

 \\hich does not vanish, is 



-pjm' (m'+ 1 ) d&S sin (r'X'-a'). 



Consequently, to order 



1> 



, and of = a, 



and no other term is of order l/d . Next, taking terms of order d,/o? , it is evident 

 that these can only arise from ~R m - T (0), R m - T (l), which involve d 1 Q m ' ,' / , C^Q - *i*'* 1 . 

 Eqtiating the sum of the coefficients of the terms containing these harmonics, with 

 factors d and d l} to zero, we get the following general expressions for p m '/, p m '\* 1 , 

 to order d t /d a : 



(5) 



T _ 



PM '~ I 



/i sin S m' (m'r'+l) 

 d a (m' 



ro'(2ro'+l) 



Pm 



i cos S 



m' 



m'+l 



V', 



' I 1 \ tCl^l I 1 \ P"' ' 



T -_, _ i d l cos 3 (m r + 1 ) (m f + r) (m' +r'-l) T 

 * ' / /( 



If the type of any of these coefficients exceeds the degree, it must be set equal to 

 zero. No other terms are of order djd^. 



So far no restriction has been placed on m' and T. In making a further approxima- 

 tion, we shall not write out general expressions, but shall consider the effect of the 

 second order terms (d.Jd and df/df) on two specific terms in the velocity potential 

 of the atmosphere, viz., Q, 1 sin (\' a,) and Q 2 3 sin (2\' 04), which give rise in ^ 

 to terms 



-Q^sinx'-a, and -' 



The terms of the proper order on the right-hand side of (4) are (a) those involving d. t 

 from R m . T '(2), and (b) those involving df/dj from K w1 ' (p) and R m . 41 T '*' (p) 

 where p = 0, 1. 



VOL. ccxill. A. 2 q 



