540 Mr Hargreaves, The Harmonic Expression of the 



| 5. Returning now to the particular case of the functions ^ 

 required for a cliathermanous atmosphere, p= 1, and we need only 



write the s suffix ; accordingly with a n = — ^t^— r^ — ^. used for 



& J 2.4... 2n 



abbreviation, 



-* ^ s^ 77 " * k (- 1 ) M ~ 1 ( 4w + !) ft » P / 2 n(sinX) P / gw (sin g ) 

 Y 1 =-C0S\C0S0+~C0SA.COSdi —7^ ,-770 — nvT~- r> /0 .tv 



^4 2 „=i (2w— l)(2?i+2) 2n(2?i+l) 



?Tcos 2 \cos 2 SS P // 2M (sinX)P // m (sing) 



%2 _ 2 cosA,coso2, „ „ {2n _ 1)2n{2n+1){2n+2) 



^ 3 = ^cos 3 X,cos 3 S2 



7T 



^ 4 = ^cos 4 A,cos 4 8S 



P ,// 2 , t (sinX)P /// 2n (sing) 

 (2w-2)(2w-l)2w(2n+l)(2n+2)(2n+3) 



P /,,, 2n (sinA.)P //,/ 2n (sinS) 



(2w-3)(2?i-2)(2w-l)2w(2w+l)(2n+2)(2n+3X2w+4); 



(21). 



First notice that all these coefficients vanish at the poles, as 

 they should do, for there is no daily variation proper at the poles. 



Again, sin \, sin 8 change signs, one in passing from northern 

 to southern hemisphere, the other in passing through the equinoxes ; 

 cos \, cos 8 do not change sign. 



Hence we see that ^ has a single even term j- cos \ cos 8, the 



remaining odd terms, % 3 % 5 ..- are entirely odd, % 2 % 4 . .. entirely even, 

 conclusions which may be drawn directly from (3). When sin 8 is 

 expressed in terms of solar longitude by the relation 



sin 8 = sin e sin 6, 



%3%5 • • • an d the odd part of Xi have the form 



«! sin 6 + a 3 sin 30 + a 5 sin 59 + ... , 

 YsXi and the even part of ^i have the form 



a + a. 2 cos 26 + a 4 cos 4$ + 



Where an odd power of cos 8 occurs, an elliptic integral is wanted 

 to complete the harmonic expression, thus the even part of % x , viz. 



j cos X cos 8 = cos \i—+E 2 cos 26 + E± cos 40 ... J 



= cos X (-7533 + -0324 cos 26 - -0003 cos 40 . . . ), 



