124 



PROFESSOK G. H. DARWIN ON THE INTEGRALS OF THE 



THE HARMONICS OF THE THIRD ORDER. 

 In these the only coefficients are A , A I} A, A s , and (2) becomes 



' COS 





sin 



snr p 



+ 2 (/C 2 - /C' 2 ) [i A A ! - A /CVM A + & lA 



+ -& (4 - 9/cV 2 ) /V 2 - yihr (12 - 25*V 2 ) KV 



A (^ ~ 



(* = 0, 1, 2, 3). 



(1) and! (4) 



These are defined thus : 



m/ (u.) = sin S (/c 2 sin 2 - 



Second Tesseral Cosine Harmonics. 



= - K cos 



2 



- /c' 2 cos 2 <)*, (s = 0, 2 



(36) 



where g 3 = f [1 + K 2 ^ (1 f K 2 + ^ 4 ) i ] 5 with the upper sign for s = and the lower 



for s = 2. Writing 



= i [3/c 2 - 2 (4 - 7/c 2 + 4/c 4 )'] = i [1 - 3/c' 2 (1 - /c' 2 + 4/c' 4 ) 4 ] 



C 3 ' ((/) = . 

 where f = - , g = 1. 



K 



Squaring ^./ we find 



- K cos 

 ^ + K' 2 sin 2 





After some rather tedioiis reductions I find (for ,s = 0, 2) 

 ' (cos) = 4 ^ 3 ? & 3 C 8 y - (t ^ - A (1 - SO + y5 (4 - 25K' 2 



olil ^J 



+ -^ (2 - 5^) K 2 K' 2 < 2 + 3 -L K v*} / 



Now writing D = (1 K /2 + 4: /4 ) i J 



5< 2 = 1 - 3/c' 2 Z>, 



5 2 * = 2 - 7/c' 2 + 13/c'+ 2(1- 3/c' 2 ) D, 

 5 3 < = 4 - 21/c' 2 + 48/c' + - 63/c' fi (4 - 19/c' 2 + 31/c'*) Z>, 



= 8 - 56/c" 2 + 177/c'* - 314/c' c + 313/c' 8 (8 - 52/c' 2 + 136/c' 4 - 156K 78 )!). 



