Mr. H. Hennessy on the Attraction of Spheroids. 25 



demonstration more easily comparable with that which he has 

 given, I shall adopt the same notation. Hence if 



V— C* /* 2 */* R p r' 2 dr'tlco' sin Q'd V 



J Q J J Vt s — 2/r'[cos0cos0' + sin0sin0'cos(a> — «/)] -f r' 2 ' 



V is a function, the differential coefficient of which with re- 

 spect to r, the radius drawn from the origin of the co-ordi- 

 nates to an attracted point, would represent the attraction of 

 any spheroidal mass on that point in the direction of r. The 

 distance from the origin of an elementary parallelopiped of 

 the spheroid is represented by i\ and the density of this pa- 

 rallelopiped by p ; 0' represents the angle between r J and any 

 one of the co-ordinate axes, and w' the angle comprised be- 

 tween the projection of r on the plane of the other two axes, 

 and the direction of either of them. The angles and w refer 

 to r in the same way that 0' and co' refer to r'. The limit 

 7r refers to 0', 2ir to «;', and R, the radius of any point on the 

 surface of the spheroid, to r'. 



2. The theorem to be proved may be thus stated. 



If a material point be situated on any part of the surface of 

 a spheroid having the radius r at that point, and if the sphe- 

 roid differ so little from a sphere having the radius a, that 

 r—a{\ +ay\y being a rational function of ju, and a>, and a a 

 constant so small that any terms in the development of r mul- 

 tiplied by its second power may be neglected, then for the 

 point in question, 



„ „ d,Y 47r .„ . 



v+2a ^=-y* w 



But* 



, T lira 3 ua s / rT TT a _- a 2 , \ . . 



- V= -3T + 7- ( U o + U,-+U 8 p r +&aJ, . (a.) 



U , Uj, &c. satisfying the equation of Laplace's coefficients. 



If, for brevity, the term containing a as a factor be represented 



by u, we shall have, after differentiating and multiplying the 



result by 2r, 



„ dV 8?ra 2 „ du 



2r -T- = — + 2r -y- ; 



dr 3r dr' 



and adding (a.) to this, 



TT „ dV 47rfl 2 M du 



v+2 ''*=-^ +tt+2r *- • • • w 



In (a.), is evidently the term in the attracting force of 



3r 



the spheroid depending upon the sphere with the radius a ; 

 * Theorie Analytique, &c.,tom. ii, p. 372. 



