No. XV. 



Solution of a General Case of the Simple Pendulum. By 

 Eugenius JSiilty. — Read s,ist August, 1818. 



In a letter which I wrote to Dr. Patterson, and which the 

 Society tliought worthy of piibhcation. 1 found a new con- 

 Terging series for determining the times of oscillation of 

 the simple pendulum in a plane. This has since led me 

 to consider a more jieneral case of the simple pendulum, 

 in which the motion is supposed to arise from the action of 

 gravity and an impulse not directed in a vertical plane, and 

 to take place in the surface of the sphere of which the radius 

 is the length of the string connecting the oscillating point 

 and centre of suspension. As the series which I have found 

 in solving this problem have not been noticed by the latest 

 writers on mechanics, I have thought that the following in- 

 vestigation might not be unworthy the attention of the So- 

 ciety. 



Let X, y, z be the vertical and horizontal distances of the 

 oscillating point from three rectangular planes X. Y, Z, given 

 in position with respect to the centre of suspension, d^ the 

 element of time during which the motion is considered as 

 uniform, and^ the accelerating force of gravity. 



The velocities at the beginning of dt in the directions of x, 



y,z&ve _?, J^, -^ ; the forces lost during this element are 

 HZ or Qv 



