254 



Curve described by a Movable Pulley. 



To resolve this 

 problem, let us as- 

 sume the equation 

 of the ellipse 



a 2 y 2 +b 2 x*-=a 2 b 2 



2 



or 



V 



x 2 



1. 



(1) 



c 



Calling the distance between the foci 2c, we have b 2 —a 2 

 ailing the angle of the tangent with the axis « and differen- 



tiating (1) we have 



dy 



tan a 



dx 



b*x 

 a 2 y 



(a-c^x 



a 2 y 



m 



(2) 



m 



We 



From equation (1) we get 



V 



b' (a 2 -c 2 ), 



from (2): 



a 



a 2 



a 2 



c 



a 



2 



my 



x 



? and a 2 



c ? :r 



These in the last equation give 



my -f x 



y 



x 2 -\-yx 



1 



m* 



m 



+ 



0, which is the equation required — (3). 



Comparing this with the general equation of lines of the sec- 



ond order 



Ay 2 + Bxy+Cx 2 +Dy f Rf + F ^ 



1 



m 2X2 



m 



+ 4>0. 



we find B 3 -4AC 



Hence the curve described by the pulley is a hyperbola whose 

 centre is the centre of the ellipse, and whose axis is inclined to 

 the axis of the ellipse. 



To determine this inclination and thereby the position of the 

 hyperbola, let m represent this angle. Then the equations of 

 transformation from one set of rectangular axes to another, the 

 origin being the same, are 



x=x' cosw-fy' sin<» and y =y' co$m — x' sin&> 



These values of x and y substituted in (3) which may be trans- 



formed into 



2 



y 



j: 



+ ^m~2^ + c2 



(because 



1 



m 2 



m 



l+tan 2 « 

 tan« 



J2_ 

 tan 2« 



will give 



„/ . 2coswsinw\ /. 2coswsint" 



y * I cos 2 &/-sm 2 «H tt — -hr sin 2 ^-cos 2 w-- 



\ tan 2« / \ ian*«* / j/^\ 



t ^ /cos 2 t»» sin 2 "> \ i 



+2*y'\;— sr - — — -2cosu,sinc*>]+c 2 =0..... J 



tan 2" 



\tan2u tan 2« 



