416 Mr. D. Vaughan on the Form of Satellites 



This expression, on substituting for / its value 



/ 2A 2 B cos <ft cos fl \ 

 VA^os^E^ 2 */' 



becomes 



or 



<£ + B 2 sin 2 </» 



2^fc 2 A 2 B cos 8 <f> cos*0d<t> d6 

 A 2 cos 2 </> + B 2 sin 2 <£ ' 



2^ 2 Bcos 8 <£cos 2 d$ dd 



l-e 2 sin 2 </> ; {Zd > 



and the attraction of the spheroid at the point in question will be 



2 9 k*B$coMS p i -™l»™* d t (24) 



The last integral, or 



J(l — sin 2 <f>) cos <f> d<f>__ sin <f> __ 1 — e 2 1 + e sin <f> 

 l-e 2 sin 2 <£ ~ " ~? 2?" S 1-esin^ 



7T IT 



which becomes, on taking <f> within the limits of + ^ and — » 

 2 l-e s . l + <= „ ' 



s-^r i0 SYzr e - • m 



Substituting this value for the last integral in (24), and making 

 a second integration, there results 



A 721I /sin20 , 6\(\ 1-e 2 1 + A 



and this, taken within the limits of 0= + g- and — ^-, becomes 



2 *VB(l-^- 2 logi±^). . . . (26) 



This expresses approximately the attractive energy of the body 

 at the extremity of its axis of rotation. The augmentation of 

 gravity which the disturbance occasions at this locality may be 

 deduced from formula (3), which, on substituting B for r, and 

 retaining only the first term of the series, becomes 



4^ 2 ttBR 8 



3s 3 ™' 



Adding this to (26), we arrive at the following equation for the 

 value of G', which shall be used to represent the force of gravity 

 at the poles of the satellite, 



A course similar to that pursued in deducing formulas (20) and 



