Curve described by a Movable Pulley. 



257 



- 



f 





the fixed pulley, a body can be raised and transported to a new- 

 position, without the use of complicated machinery. 



For this purpose let the body rest at C, and suppose we wish to 

 transport it to m. Regarding the pulley and body as points, the 

 question is reduced to finding an equilateral hyperbola passing 

 through the points C, m, and B. The angle <*, c, and the position 

 of the centre must be determined with regard to the point C. 



Let u and v be the coordinates of the point C : and through 

 this point take two rectangular axes ex', cy', one parallel to the 

 horizon and the other perpendicular to it ; u and v will also be 

 the coordinates of the centre O of the hyperbola relatively to 

 point C. Since the axis of the equilateral hyperbola (see (5)) 

 makes an angle of 45° with the horizon, the equations of trans- 

 formation to the point C will be 



y=t>-(y.-*,) ^i 



by means of which, 

 (5) becomes 



u2 -v 2 -\/2(u + v)x l 

 -\/2(u-v)y [ +2x l y l 



=-c 2 sin2«. 



For the point Cx l 

 and 



4. 







y , ~ 0, which re- 



duce this to 



= c 2 sin 2a. 



whilst, 



u 



(6) 



v 



-vf2(u- V )y,=0. 



it wand n denote 



the coordinates of the 



point m we shall have for this point 



mn\/2—{u-\-v)m-(u — v)n=0. 

 We may also take as known the position of the point B of the 

 curve, and denoting its coordinates by p and q we have 



pq V% - (« + v) p - ( u - v) q =0, 



whence u _^{ p - q) - pq(m -n) 7nv{p+q)-pq{m±n) 



nence u _ ^ £ j » - ^ 2 ( mq - np) 



from these equations we know the value of c 2 sin 2<* : but c and « 

 remain yet undetermined to the great advantage of practice ; be- 

 cause we can choose the inclination or the length of the line AB, 

 according to convenience, provided they satisfy equation (6). 



Substituting in (6) the values of u and v just found, the con- 

 dition is reduced to 



. • « o (n-q)(P_z™} 



c- sm2a=;2mnpq-^l mp y 



the second member of which is entirely known ; and any value 

 assumed for c or «, will give the corresponding value of the other. 



Second Series, Vol. VIII, No. 2 



8. 



Sept. 



1849. 



33 



