RESULTING F K () M T II K TIIKORY OF THE GYROSCOPE. 



To refer tin- ili.^i/n'-i UK ,i> e\preN..,-d hy (:>(>) to the polo of the ecliptic 0, as 

 angular motion, \\< must (see note pa^e N) multiply by cos J/'/-' or (from which it 

 dittt TS hut slightly) cos i and divide 1>> sin /; \vr thus obtain for the elementary 



j,l" ' *lttll 



jf 



:;^ cos MM' sin i cos i dt, 



sin / 



and for nutation 

 :}</. A' MII J/J/' sin / (//. 



The maximum value of the arc MM 1 is about 11 56'; the line MO describing 

 during an entire revolution of the moon's nodes an elliptical cone about OS of 

 which the minor semi-diameter (Xl/') is 5 8^' (about), and the semi-major 11 56'. 

 Pi\ conceiving the elementary motion of the pole of the earth (or its gyration) as 

 at each instant about the line J/0, as it makes its conical revolution, the undulating 

 nature of that motion, or the "nutation," is easily conceived. 



Approximate values of sine and cosine of MAF may be determined from (35); 

 which, substituted with those oft (3*2) in (38) and (39), will enable us to integrate 

 and obtain very accurate expressions for the lunar precession and nutation.' 1 ' But 

 these expressions, nearly free from errors of approximation, may be more elegantly 

 determined as follows: When the angle of the moon's orbit with the equator is 

 minimum, the angle i=If; when maximum i=I-\-f (epochs corresponding to 

 ii z f=0 and s /=7i); the angle MM' is zero, and the corresponding rates of pre- 

 cession are by (38) 



When ./=^7r, we have 



sin 1 \r*r cos /sin /' 



MM - sin MM= 



V Icos 1 / cos- /'' V 1 cos 2 / cos- /' 



cos ATF=cos /cos 1 f, cos i=cos /cos F, sin i=l/l c s " / ( ' os " /' (the three 

 first being the residuals of exact analytical expressions after omission of quantities 

 of inappreciable magnitude). 



Hence, by (38), the rate of precession for n,f=\-x. is 



(c) K cos /cos 2 / 1 * 



Assume the formula for precession to be 



K(Pt+P sin nJ+P" sin 



* When the moon's orbit intersects the ecliptic in a solstitial line, the elementary procession IT cos i 

 nboul UK own pole is reduced, with but slight error, to this same about the ecliptic pole, by 

 simple multiplication by cog /' : a result coinciding with the above. 

 '" Sre Additional Notes, p. 51. 



