APPENDIX. 



VU 



(T0 = 





■(T). 



Having determined the periodic times of a particle, or of an 

 insensible mass, by the aid of this, we may next ascertain the 

 periodic time at an assumed distance E A, situated as in Fig. 1, 

 by the application of Kepler's third law : let this be represented 

 by (0- 



Now let the attractive forces of the moon and the earth, acting 

 at A, be separately computed in accordance with the law of trra- 



. . / M \ ^ 



vitation y^'^J, and then, taking the difference of the two forces, 



let the same be expressed in terms of earth's force F, and let it 

 be denoted by ~ F, and (f) represent the periodic time due to '^ F 

 at the same distance E A. We shall have 



O'y 



F 



If (ty thus ascertained is found to agree with the moon's peri- 

 odic time, the point A is well determined. Jf there is any differ- 

 ence, that may be made to disappear by the application of the 

 method of trial and error. When A is situated beyond the moon, 

 the sum of the attractive forces must enter instead of their differ- 

 ence; and so also in the determination of the point B. 



The positions of the points A and B, on the hypothesis that 

 tlie girdle on the one side is between the earth and the moon, 

 may be learned from the following table: — 



Moon's distance 



In Perigee 

 56.964 



Meau Dist. 

 60.273 



In Apogee 

 63.5831 



Int. dist. 

 Ext. dist. 



E A 

 EB 



48.309 

 56.190 



51.116 

 60.090 



53 922i 

 63.389 



On the hypothesis that the girdle encompasses the moon, as in 

 Fig. 2 : — 



Dist. Ext. 



In Perigee 

 66.426 



Mean Dist. 

 70.285 



In Apogee 

 74.1441 



The distances are all expressed in terms of the radius of the 

 eartli's equator. 



The position of the points in question ultimately depends upon 

 the ratio of the masses ; so that the E A and E B of the girdle 

 in apogee, perigee, and mean distance, respectively, preserve their 

 ratios to the moon's own distances, and hence A and B move 

 in ellipses similar to the moon's own orbit; the girdle at those 

 places expanding and contracting. The self-adjusting material 



(25) 



