﻿518 



On Secondary Planets in Small Orbits. 



being supposed to increase from 58° by uniform increments. In 

 taking c equal to sin 6, 6 in the case in question cannot differ 

 much from 90°, this being its value for a prolate spheroid. I 

 have therefore deemed it best to assume for 6 the constant value 



of 80° until the results exhibit a maximum value for -^-^: and then 



D d 



we may determine the extent to which the arc must be altered 



in order to fulfil the other conditions. Putting P', Q', and B/ 



for P, Q, and R, divided by 47r# 2 $A 2 , the relation between these 



r 3 

 quantities, and also the values of ^ for different amplitudes and 



moduli, will be as follows : — 



0, 



0. 



P'A. 



Q'B. 



R'C. 



r 3 

 D3* 



o 



80 



o 



58 



•1883131 



•1201880 



■1163499 



•0659336 



80 



59 



•1830151 



•1147750 



•1107895 



•0663581 



80 



60 



•1776240 



•1093840 



•1054360 



•0666343 



80 



61 



•1721416 



•1040776 



•0998855 



•0669875 



80 



62 



•1665670 



•0985860 



•0944249 



•0672047 



80 



63 



•1609083 



•0935796 



•0890777 



•0671888 



80 



64 



•1550663 



•0884503 



•0838432 



•0669370 



80 



65 



•1493456 



0834122 



•0786740 



•0667006 



80 



66 



•1434507 



•0784623 



•0735980 



•0662014 



79 



62 



•1674540 



•1000739 



•0948294 



•0676545 



78 



62 



•1684030 



•1014850 



•0952608 



•0681364 



From the above Table it appears that r^- 3 reaches its highest 



limit when cj> is 60 degrees ; but to fulfil with this value equation 

 (25), 6 must be reduced to 79 degrees. From the amplitude and 

 modulus thus obtained it may be easily found that 



B = -4988A, C = -4695A, and ?=2'4547. 



r 



It thus appears that a homogeneous fluid satellite as dense as its 



primary would be unstable, even in a circular orbit, if the radius 



of the latter were less than 2*4547 times that of the central orb, 



and that, when near the verge of instability, the principal axes 



would be nearly in the ratio of 1000, 499, and 469. At' the 



point of its surface nearest to the primary the attraction would 



be almost equal to half that of a sphere with a diameter equal to 



the major axis, and it would be reduced about 40 per cent, by 



the disturbing force of the central body. 



When the primary and secondary are unequally dense, the 



formula for the least distance between them must be modified by 



regarding r as the radius of a sphere equal to the central orb in 



mass and to the satellite in density. Now r is equal to r'^/p, 



