SIMPLE PENDDLUM. KiM 



dt ^ ' 



The equation (3) expresses the known principle of living 

 forces, and the equation (4) that of the equable description 

 of areas. 



Let V be the horizontal angle formed by the projection of 

 a on the plane X, and by the vertical plane Z. Then it is 

 evident that y=\/{-zax — .T').sin. v and z=>/{'Zax — x').cos. v ; 

 and their differentials are 



dy-—p ^.s\n.vdx+x/(s,ax — r')cos.T'dtJ, 



•^ vinax—x") ^ ^ 



dz= — ^.cos.rd.r — ^(2ax — :i;*)sin.7)dr. 



^(200: — X-) ^ ^ 



Substituting these expressions in the equations (3) and (4), 

 we shall find after ol)vious reduction, 



i^p ^ — ^2s(zax—x-).(b—x), 



(2ax — a:')dT;_,, 

 d^ 



from the first of which eliminating dv, and from the second 

 d^, we get 



dx C2ff)* r. /-X 



_ = ±(^^f. (5) 



in which F=x' — (2a + b)x^ + 2af -r — c, and c = — . 



These are the equations from which the motion of the pen 



VOL. II.— 3 P 



