RESULTING FROM T II K T1IKORY OF T 11 K (i Y R O SCOPE. 







being a revolution about an axis (the pole of (In- ecliptic) perpendicular to the 



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



.) i 



(',':}) and make /= *' , and =i*, we shall get 



"i > 



., , C A . r 



- N - :* n ., sin /; 



n C 



and this will he the total angular motion of the pole P about the solstitial line TO 

 in <>nr >> i-nl iiiiini <>f tti> KIIH ; but by tliis very motion of the pole the equinoxes 

 have moved an angle measured by this </!NJ,/<H-I nn at referred to the pete of the 

 i<'lll>tii- that is, by the angle expressed by ('21) and the solstitial line T C has of 

 course, undergone tlie same imnement. and the next annual gyration will be about 

 the consecutive line 7" < ', and so on; producing a continuous motion of the pole 



/'about the pole of the ecliptic P". 



To obtain the pr< -cession due to the moon, it is necessary to substitute in (19) for 



S I/' 

 , ' , in which M' is the attractive force of the moon and (/) its mean distance. 



* ' 



* I 



IJut '/''(time of moon's revolution) is, by Kepler's laws, ,,,'_;_ ,,.. = r ,i_i STJ!T|~~ 



(calling the mean angular velocity of the moon n t and the ratio of earth's mass 



M l n * 



to that of moon's mass, ,-) ('2>S<t); hence = * 



(r) 3 



If t 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, />/;/"/ to tlie pole of the moon's orbit, expressions analogous to (25) 

 and ('->('>). 



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

 double that of the sun, yet the larger divisor 2 , introduced by integration, renders 

 the value; of ('.'(I) and of the fluctuating term of (25) very small for the moon say 

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

 disregarded in the lunar expressions. 



The elementary precession due to the moon about (fie pole of its oicn orbit would 

 be by (25) 



29. "*' ^T- 



C 



From this, by the usual methods, can be deduced the 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 gyration produced by the moon directly to precession 

 and nutation. 



If we substitute for ?> , " 2 , and sin t for sin /, in (28) we shall get, for the 



A-N 



total gyration nbmtt the line of grenlrnt declination, produced by one revolution of the 

 moon in its orbit, the expression : 



30. 3 *- -v* r "T 4 sin/. 



2 December, 1871. 



