﻿in Small Orbits. 517 



A 2 — B 2 

 in which the modulus c 2 is equal to -p — ™, the limits of <f> are 



n 



from to cos~ l <£= -r, while n is put for \/A 2 — C 2 , 6 for 



\/l — c 2 , and A for V Y — c 2 sin 2 <£. If the above values of P and 

 R be substituted in equation (21) , there will be obtained, on di- 

 viding by Anrl&g, 



' ^[F(*.0-B( e .0]-^[Atan*-E(e.0] 



= Fa( A2 +?) • <™ 



Let the ratio of the principal axes be now expressed in terms of 



the modulus c 2 and the amplitude <j>. As cos cf> is equal to -^-, 



A 2 — B 2 

 and c 2 to A2 _n 2 ? it may be readily found that n 3 is equal to 



A 3 sin 3 <f>, and B equal to \/l — c 2 sin 2 <£ or to A. Substituting 

 these values for B, C, and n 3 in the last equation and dividing 

 by A 2 , we obtain 



^[»(..«-i(»:#n-i£*[At»#-i(...«] 



= ^ 3 (l + ico S ^) (24) 



A substitution of the values of R and Q in equation (19) will* 

 by a like process, lead to the following : 



A 3 cos<£ r_ , ,. c 2 sin$cos<£ , 2T?/ 1 r 3 2 

 _^|E(c.« U^ 2F(c . 0) J = __ cos ^ 



For finding the greatest extent to which the satellite could 

 have its form elongated and its orbit reduced, I have computed 



the values of ^3-, and of the relation between the forces along 



the principal axes for different moduli and amplitudes, the latter 

 Phil. Mag. S. 4. No. 276. Suppl. Vol. 41. 2 M 



