414 Mr. D. Vaughau on the Form of Satellites 



excess of the major axis over either of them. They may there- 

 fore be considered equal, at least in the first approximation ; and 

 if greater accuracy be required, it may be obtained by introdu- 

 cing corrections depending on the square, the fourth, and the 

 sixth powers of the excentricity of the elliptical section in which 

 both arc situated. Accordingly, let the satellite be regarded as 

 a homogeneous prolate spheroid ; and let A and B represent the 

 major and minor semiaxes, the former of which always ranges 

 with the primary's centre. To find the attraction at the extre- 

 mity of the major axis, conceive the body to be divided into 

 innumerable sections by planes passing through this line, while 

 these sections are subdivided into a corresponding number of 

 infinitely small pyramids, having their vertices all terminating 

 at the proposed point, which shall be taken as the origin of the 

 coordinates. Put for the angle which any section makes with 

 the plane of the orbit, / for the length of any of its pyramids 

 whose inclination to the major axis is denoted by <f>. The vertical 

 angles of this minute pyramid will be d<f> and dd sin <f>, and the 

 component of its attraction in the direction of the major axis is 



gkH sin </> cos <f) d<f> dd (12) 



Substituting for / its value . 2 . g ? — g-r, this expression 



becomes 



2gh*AB* sin <f> cos 2 <f> dcf> dO 2^ 2 B 2 sin cf> cos 2 </> d6 dd 



A 2 sin*(/> + B 2 cos 2 </> ° r A(l-e 2 cos 2 <£) ; " 6 > 



and the double integral of the last quantity, or 



^HtSS? <»> 



taken within the proper limits, will represent the attraction of 

 the satellite at the extremity of the major axis. Now 



J cos 2 (f> sin <j> dcf> __ cos </> 1 , l + ecos<£ ~ ._ R . 

 l-e 2 cos 2 tf> -"l 2 ~"""2? 10g l-€ 2 cos</, +L ' ' (16j 



IT 



When this is taken within the limits of </> = 6 and <j> = — , and sub- 

 stituted in (14), the latter becomes 



and making a second integration between the limits of 0— +w 

 and 0= — 7r, the expression for the attractive force is 



4^ 2 ttB 2 / 1 . 1 + e 1 \ nm 



-^W> g i^ € -?> • • • (17) 



