traced upon the Surface of the Sphere. 



331 



Let EQ be the equator, P its pole ; SB, the ecliptic, and M its pole. 

 Refer the whole system to the equator and the meridian through the tropic 

 of Capricorn. Then S in PM is that tropic, and PM is the measure of the 

 inclination A of the ecliptic to the equator. 



FIG. 23. 



The equation of the ecliptic is, then, if SPR = 0, and PR = (j>, 

 cot</>, = tan \cos6, , (1.) 



But if SR = x = SMR, we have 



. cos SR = cos SP cos PR + sin SP sin PR cos SPR, or 



(7T "\. S 7T \ 



-0- + A 1 cos (p t + sin f + X 1 sin (f), cos 0, , 



or 



cosx = cos <p, sin A + cos A sin <p t cos 6, (2.) 



or x = cos~ 1 { cos^, sin A 4- cos A sin<p, cos 6,} (3.) 



But in the present case, if cf>, 6 be the co-ordinates of the curve sought, 

 we have < = <,, and hence from (l) we have cos0,= cot^cotA,, and 

 this put in (3), gives 



X = cos 1 { cos (f) sin A cos A sin <p cot <p cot A} 

 , f cos (b ) 



= 008-' 1 ^-|- [ (4.) 



(. sin A J 



But if n express the ratio of the earth's angular velocity upon its axis 

 to its angular velocity about the sun, we shall have 6 = n %, which, inserted 

 in (4), gives us 



