' ^^Mm of a freely suspended Pendulum. 135 



is very considerable in modifying the angular motion^ we send 

 you a complete investigation of its effects, requesting the favour 

 of its insertion in your valuable Journal. 



We remain, Gentlemen, 



Yours, &c., 



Joseph A. Galbraith, 

 Trinity College, Dublin, Samuel Haughton. 



"July 14, 1851. 



If a point P move 

 on the surface of a 

 sphere under the influ- 

 ence of a force F, which 

 acts in the tangent to 

 the great circle joining 

 P with a fixed point C 

 on the sphere, it will 

 describe a spherical el- 

 lipse roundC as centre; 

 if the force F act from 

 P towards C, and be 

 equal to g tan r sec^ r, 

 r being the angle at 

 the centre subtended by the arc CP, and g the accelerating force 

 of gravity*. 



Let o) be the angle which CP makes with the axis, m the 

 angle which the semidiameterCM, conjugate to CP, subtends at 

 the centre, cJ the angle which CM makes with the axis : let also 

 a and h be the semiaxes major and minor, a and /3 the tangents 

 of the angles which they subtend at the centre, v the velocity of P, 

 and p the perpendicular arc, drawn from C to the tangent. 

 The following fundamental equations connect the motion of P 

 with the elements of the ellipse : 

 ^ F=^tanr(l+ tan^r) 



sin^ rd(o =^ \f g a^ dt 



„- ^^«^ I 



sin^ J 



Let a small force R acting in the direction of the tangent to 

 CP at P, and outwards, at each instant disturb this elliptic motion^ 

 we may still suppose the point P to move in an ellipse, the mag- 



* This elegant theorem is due to Professor Graves, who communicated 

 it, together with some others connected with the motion of a point on a 

 sphere, to the Royal Irish Academy, January 24, 1842. 



L2 



(1.) 



