RESULTING FROM THE THEORY OF THE GYROSCOPE. H 



To refer the displacement expressed by (36) to the pole of the ecUptic 0, as 

 angukir motion, we must (see note page 8) multiply by cos M'F or (from which it 

 differs but slightly) cos i and divide by sin /; we thus obtain for the elementary 

 precession 



38 — ^*^ cos MM' sin i cos i d(, 



sin/ 



and for nutation 



3i?. K sin MM' sin i dt. 



The maximum value of the arc MM' 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 {SM') is 5° 8^' (about), and the semi-major 11° 56'. 

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

 at each instant about the line MO, 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 MM' may be determined from (35); 

 wliich, substituted with those of i (32) in (38) and (39), will enable us to integrate 

 and obtain very accurate expressions for the lunar precession and nutation.'*' 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 /=/—/'; when maximum /=/-(-/' (epochs corresponding to 

 vj^=^ and n^f =7c); the angle MM' is zero, and the corresponding rates of pre- 

 cession are by (38) 



(a) A-?^?^Pfor;V=0. 



2 sur / 



(6) 



When Vjt^^^n, we have 



j^sin 2(14-1') f , 



K ^ — ' — -' tor iiJ=7t. 



2 sin / ' 



,,„, sin 1 ■ ,,,,- cos /sin /' 



cos MM =1 - , sm MM- 



V 1 — cos'7 cos"' /' V 1 — cos' 7 cos' r 



cos if i^^cos / cos- /', COS i'=cos /cos /', sin /=l/l— cos' /cos' /' (the three 

 first being the residuals of exact analytical expressions after omission of c^uantities 

 of inappreciable magnitude). 



Hence, by (38), the rate of precession for n^t^^^n is 



(c) K cos / cos' /' * 



Assume the formula for precession to be 



Z(/^+P'sin «3<+P" sin 2n,t); 



* When the moon's orbit intersects the ecliptic in a solstitial line, the element;! ry procession Kcos i 

 (29) about its own pole is reduced, with but slight error, to th(' same about the ecliptic pole, by 

 simple ranltiplication by cos /' : a result coinciding with the above. 



"' See Additional Notes, p. 51. 



