Curve described by a Movable Pulley. 



255 



which will be the equation of the hyperbola referred to its centre 

 and axes, if 



cos 2 m 



tan 2a 



sin 2 w 

 tan 2a 



sin oi cos w = 0. 



or (clearing fractions and dividing by cos 2 o>) if 



tan 8 w -f. 2 tan 2«. tan « — , 1 * 0, whence 



tan w 



tan2«:fcv / l + tan*2« 



d=l — sin 2« 



tan 2a ± sec 2a, or 



taiiw 



cos 2« 



but 



# 1 4- sin 2a 



f cos 2a 



sin(90°-[45 



2cos 2 (45°-a) 



cos (45 



«) 



2cos(45°+«)cos(45°-«) 



cos(45°+«) 



o 



«]) 



COS(45°+a) 



sin(45°+«) 

 cos(45°+«) 



tan (4 



5°+«) 



in a similar manner we find the first value of tan « 



tan & = -f- 



tan w 



a* 



1 — sin 2« 



cos 2« 



tan (45 

 45° - «. 



4- tan ( 45° — « ). Hence 

 a). Therefore 



or 



180 



(45° + a). 



The difference between these two values is 90°. Therefore if 

 the first value is the inclination of the transverse, the second is 

 that of the conjugate axis. 



Since « is the angle made by the axis of the ellipse with the ho- 

 rizon, and w that made by the axis of the hyperbola with that of the 

 ellipse, (w-f a) is the inclination of the axis of the hyperbola to the 



~ ' " =45°: therefore its axis makes 



8. 



M 



horizon. But <■» + « --=(45° -«)-*-« 



an angle of 45° with the horizon, and is entirely independent of 



the inclination of the line of the foci to it — which is remarkable. 



It may also be shown that this 

 hyperbola is equilateral, and con- 

 sequently the vertical line MOM' 

 and the horizontal POP' are its 

 asymptotes; because, observ- 

 ing that sin 2 ci 



2sinwcosw=sin2..*, and substi- 



COS 2 m 



cos 2« ; 



tming for ci its value (45°-«), 

 the coefficient of &* will be re- 



ducedto-eos2 





cos 2a 



tan 2 



sin 2 

 tan 2a 



sin 2« 



</. 



i 



sin 3 2«+cos 3 2« 



sin2« 



1 



sin 2,,' and that of V'\ to +sx^2« 



Sin (45^+a)=vl (cos a+ sin a); squaring, 2 sin 2 (45°+4= ^ a+sin 2 a+2 sin a 

 l a==rl+sin2a - Hence l4-^m2aW > ^i- J (45°+a>=2cos2(45 -a). 

 T 2cos (45°-f>r2)cos(45 --a) = 2(sin2 45°sin2a-< ft* 45°COs2 a)=2cns2 45°(sin2 a 



-cos2a)^ 2 ( v /|)2 C os2a = cos2a. Hence cos 2a= 2 cos (45°+*) eos(45°-a). 



