56 STATEMENT AND EXPOSITION OF 



The middle line of the girdle will, notwithstanding, form an oval, which, at any 

 time, in its arrangement around the earth, will not anywhere be fomid at a dis- 

 tance differing much from that of the moon at that time; except in those portions 

 comparatively near to the moon. 



That part of the oval nearest the moon may pass between the moon and the 

 earth, as in Fig. 13; or else outside of the moon, as in Fig. 14; in both of which 

 E marks the position of the earth, and ilf that of the moon. 



In the determination of the dimensions in either case, it will be convenient to 

 ascertain the periodic time of a particle, or of an inappreciable mass, revolving 

 around the earth at the mean distance of the moon; which we may obtain by the 

 aid of the following formula, in which {T) will be put for the periodic time; M 

 and m representing the masses in question, and r the radius-vector ; and we have 



Then, when w is insensible, 



and when r is the same for both, from these we also have, 



(T) yW+Vn (3).,, 



(70 ^M ^ ' 



rT) = VE±^(T) (4); 



which, otherwise expressed, is 



(n = J^i-^'-m (5). 



Then, ls<— Making special application of either Eq. (4), or Eq. (5) to the exam- 

 ple in which M and* m, respectively, represent the masses of the earth and the 

 moon, and {T) the moon's periodic time, we shall have the periodic time of a 

 particle, or of an insensible mass, revolving around the earth at the distance of the 



moon. 



2d. Ascertain the periodic time (t) of the same insensible mass, revolving about 

 the earth, at the assumed distance EA, by the application of Kepler's M Lav}. 

 2>d. The attractive forces of the moon and the earth, respectively, acting at A 



may be separately computed in accordance with the law of gravitation ^--^j, and 



then taking the difference of the two forces, Avhen the state of things is that repre- 

 sented in Fig. 13; and expressing this difference in terms of the earth's force F, 



viz. as-^J^; then (with {t), the periodic time around the earth of an insensible mass 



revolving at distance IE A, already computed), we shall have 



' EncyclopEedia Metropolitana — Physical Astronomy, Section V. 



