250 NEWTON S PRINCIPIA. 



the decrement of the arc varies inversely as the weight 

 of the pendulum. 



It will be inconvenient to constrain the particle to move 

 in a cycloid : let us examine what errors would be in 

 troduced by making the particle vibrate in a circle. 



Here the force is g sin j instead of - S) hence we must 

 put 



/&amp;lt;/ . &amp;gt; u 



= I s g sin. - = --?- . 5 3 , nearly 



= ~jr 6 sin. 3 (w t -f ^}. 



Substituting we have 



#j a = 



0, a = I 



b h - gar * \ 



1 b ~~ rein* J 



the integrations being performed from nt + I = o to it. 

 Hence the arc is unaltered and the time increased by a 



II a * 

 quantity ^_:= . 



Let us now consider the con 

 struction by which Newton re- 

 I{ presented the resistance. Let a 

 straight line B be drawn equal to the arc of the cycloid 



which an oscillating body describes, and at each of its 



^ i 



points D draw the perpendicular D K equal to ~ part of 



the resistance at D, then and a l being the arcs de 

 scribed in the descent and subsequent ascent, then 



( a \ ~ ft o) a * ^ = area of curvc a K B- 



