RESULTING FROM THE THEORY OF THE GYROSCOPE. 9 



being a revolution about an axis (the pole of the ecliptic) pei-penclicular to the 

 plane of that angular motion. In other words, if wc integrate directly equation 



(23) and make ;^=— ^, and ' :=n{', we shall get 



28. 3"i7t^^sin/; 



and this will be the total angular motion of the pole P about the solstitial line TC 

 in one revolution of ilie sun ; but by this very motion of the pole the equinoxes 

 have moved an angle measured by this displacev-tent referred to the pole of the 

 ecliptic— th.^i is, by the angle expressed by (27) — and the solstitial line 7'Chas of 

 course, undergone the same movement, and the next annual gyration will be about 

 the consecutive line T C, and so on; producing a continuous motion of the pole 

 P about the pole of the ecliptic P". 



To obtain the precession due to the moon, it is necessary to substitute in (19) for 



- , ^^ , in which HP is the attractive force of the moon and (/•) its mean distance. 



But T^ (time of moon's revolution) is, bv Kepler's laws, ^n'-. r'Yr==FM -?- >: iTttT = — 



(calling the mean angular velocity of the moon n^ and the ratio of earth's mass 



to that of moon's mass, >;) (28a); hence = — '~. 



{rf l-\-yi 



If i is the inclination of the moon's orbit to the equator during any one revolu- 

 tion (regarded as constant for that time), we should obtain for the precession and 

 nutation, referred to the pole of tlic moon^s orbit, expressions analogous to (25) 

 and (26). 



Although the moon's disturbing effect, as above- expressed, is almost exactly 

 double that of the sun, yet the larger divisor ?u, introduced by integration, renders 

 the value of (26) and of the fluctuating term of (2-3) very small for the moon — say 

 about ith the corresponding values for the sun. Hence these terms are usually 

 disregarded in the lunar expressions. 



The elementary precession due to the moon o.lout the pole of its ovn orbit would 

 be by (25) 



on 8 "^' C—A . J, 



29. -^ — —^ — ^ _ — cos t dt. 



2ji(1+57) C 



From this, by the usual methods, can be deduced tlic real precession and nuta- 

 tion. But it will be more in harmony with the object of this paper, and indeed 

 more elegant, to reduce the (juration produced by the moon directly to precession 

 and nutation. 



If we substitute for 7>,, '"- , and sin )" for sin /, in (28) wc shall get, for the 



total gyration ahovt the line of greatest dedination, produced by one revolution of the 

 moon in its orbit, the expression : 



30. 3 -.^-.Tt^-sini. 



n(l+>7) G 



2 December, 1871. 



