NEWTON S PRINCIPIA. 243 



d 2 s 

 But this is also in , hence we have 



This equation being of a standard form, we can write 

 down its integral, 



= L cos A /f (t - A) 4- 

 / \/ / v mg 



where L and A are constants depending on the initial con 

 ditions of motion. 



/ f 



Take a point O in the arc, so that CO = , and let sf be 



mf 



the arc when measured from this point. Then 

 / = L cos (t - A). 



Suppose the particle began its descent from D, then A 

 is clearly the time at which the particle was at D, and 

 L = arc O D. It is manifest the greatest velocity will be 

 at O. If the particle had been undisturbed by /, the same 

 equation would have given the arc measured from C. 



If then the particle when resisted by f be at P at any 

 time, and if another particle not resisted by f be at Q at 

 the same time, then DP bears to PQ the constant ratio 

 DO to DC. 



The effect of any constant resistance is to dimmish the 

 arc continually in an arithmetical progression, but not to 

 affect the time of oscillation. 



R2 



