4,88 Mr. J. Prescott on the 



Therefore sin 6 = v /-|l — sin 2 « sin 2 <£} 



f= 1— \ sin 2 a sin 2 <£ + 1 sin 4 a sin 4 </> 



nearly, since sin a. = sin 23J°= *4 about, so that the terms 

 ignored are small. 



Hence the couple exerted by the sun 



= |- 7rBn 2 sin a sin <£{ 1 — J sin 2 a sin 2 -f | sin 4 a sin 4 $}. 



The axis of this couple is OQ, perpendicular to the plane 

 containing the earth's axis and the sun. The only effective 

 component in precession is that parallel to OP, the inter- 

 section o£ the plane of the equator and the ecliptic. The 

 other component causes nutation. Hence the effective couple 

 for precession 



= moment of couple x cos yjr. 



By spherical trigonometry 



, sin<£ 



COS ^~V(l + tan 2 «cos 2 <£) 



= sin <£(1 — J tan 2 a cos 2 cf> + 1 tan 4 a cos 4 <£) . 



Thus the effective couple is 



8tt 

 L = -r~B?i 2 sin a sin 2 <£{1 — i sin 2 a. sin 2 <f> — J tan 2 a cos 2 



+ g sin 4 a sin 4 <£ + J tan 2 « sin 2 a sin 2 (f> cos 2 9 



+ J tan 4 a cos 4 <£}. 



Now -3~=n. Therefore dt= — ^„ Hence the moment of 

 momentum generated by the sun's couple action in one year 

 d<j> 



27T 



n 



8^ . f\ 1 . .'- 3 1 ". 1, 2 



= - B?i7r sm«.7r < 1 — - sin- a . -- . - . 2 — ~ tan a . 

 5 l 2 4 2 4 



1 1 

 4'2 



3 . , 5 3 1 9 . 1, , . 2 1 3 1 

 + 8 Sm "6'4-2- 2+ 4 tair " Sma 2-4-2' 2 



l 3 / 4 * 3 X O ) 



which becomes, on putting 23^° for «, 



|Bn7r 2 sin4'y31}, 



