F. E. Nijpher — Surfaces of the Compound Pendulum. 23 



tion, while others tend to increase it. These two tendencies 

 always balance, although the value of each continually varies. 

 These two groups of particles are separated by a surface, each 

 particle of which has no tendenc}^ to change the acceleration of 

 the system, at that instant. The axis of oscillation always lies in 

 this surface. On either side of this neutral surface there must be 

 surfaces of equal tendency, those on one side having a plus, 

 and those on the other side a minus sign. It is required to 

 find the loci of these isoclynamic surfaces at any given instant. 

 This can be done by means of well known equations for the 

 pendulum, which are first given. 



In fig. 1, let represent the axis of 

 oscillation, Gr the center of gravity, and 

 S the axis of suspension. Call S G = K; 

 S = 1, and let r be the distance of any 

 element of mass dm from the axis S. 

 Let 6=ihe angle V S O, and a the angle 

 between the lines I and r, V S being -the 

 vertical plane containing the axis S. 



The entire mass of the pendulum may 

 be supposed condensed on the vertical 

 plane passing through Gr, and at right 

 angles to the axes and S, each element 

 of mass being supposed to be projected 

 along a line parallel to those axes. The 

 pendulum then becomes a thin plate 

 of varying density, lying in the plane 

 of the paper as in fig. 1. 



This supposed condensation is really unnecessary in a rigid 

 system, as the center of gravity Gr, and the element dm, may 

 lie in different planes, at right angles to the axis S without in 

 any way changing the result. 



At' any instant the linear acceleration of is g sin d and its 



angular acceleration is -y- sin d. This is also the angular accel- 

 eration of every other particle in the system. The linear accel- 



eration of d m is therefore — g sin d. 

 produce this acceleration on d m is 



~F'=dm — g sin 6 



t 



The moment of this force about S is 



r 2 

 FV = dm — g sin 



The force required to 



(1) 



If the element dm were disconnected from the system, its 



