106 Professor Airy on the Disttirbances of Pendulums, 



If I be the length of a peiKhilum vibrathig in a cycloidal 

 arc, X its distance at any instant from the position of rest, the 

 equation of its motion is 



^!=-«^. „rp„.ti,„». = f. £?*"- = »■ , 

 The sohition of this equation is 



X — a sin nt + b, 



a and b being constants depending on the length of the arc 

 and the time of passing the lowest position. The velocity at 

 any time 



— -r- = ?ia cos nf + o. 

 at 



Suppose now that, besides the force which varies as the 

 distance, another very small accelerating force / acts on the ball 

 of the pendulum in the direction in which x is nieasnred positive : 



,, d-x gx ,. d'x ,, 



The solution of this equation may still be assumed = a . sin nt + b, 

 provided we consider a and b as functions of t. For whatever 

 the solution may be, a . sm nt + b may be made equal to it by 

 assuming either for a or for b a proper form. Since then a single 

 assumption will satisfy this condition, and since we have two 

 quantities whose forms are to be determined, it follows that we 

 are at liberty to make another assumption. Let this be that 

 the velocity shall be expressed by the same form as before, 

 namely, na .coh nt + b. The convenience of this assumption we 

 .shall soon discover. 



