Curve described by a Movable Pulley. 



253 





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i. 



uncertain whether any one has determined what kind of a curve 

 is formed. This I discovered almost three years ago. and the re- 

 sult then seemed so simple and evident that I could scarcely be- 

 lieve that no one had ascertained it before : and for this reason I 

 delayed its publication. If the subject has ever been touched by 

 any one, it has never I think been considered on so large a scale 

 ns this. In the process of the solution it will be seen that besides 

 the advantage of a mechanical invention, we have in it a hitherto 

 unknown affinity between the ellipse and hyperbola and a new 

 kind of a general mathematical problem. 



Problem. — To find the curve described by a movable pulley, 

 when the fixed point and fixed pulley are not in the same hori- 

 zontal line. 



Suppose for the sake of simplicity, that both pulleys have an 

 infinitely small radius ; so that they may be regarded as points, 

 or what is the same that the pulleys are rings, through which the 

 cord passes without friction. 



Let A be the fixed pulley, 

 B the fixed point and C the 

 movable pulley. BCA is 

 the cord that suspends the 

 pulley. It is well known 

 by the principles of me- 

 chanics—that a pulley or 

 ring, thus suspended, will ""\ 

 be in equilibrio, when the 



direction of gravity is nor- 



wo/ to the curve it would 

 describe if moved freely along the cord, and always in the same 

 vertical plane. But this curve is an ellipse whose foci are A and 

 B, and whose transverse axis =AC+CB; and the line of gravity 

 being vertical, and the lowest point of the ellipse being the only 

 one at which the tangent is horizontal, it follows that the pulley 

 Will be in equilibrio only at the lowest point C of the ellipse. 



When the cord is pulled, the movable pulley passes successively 

 through other positions of equilibrium which are at the lowest 

 Points of ellipses whose foci are A and B, and whose transverse 

 axes are the lengths of the thread from A (through the successive 

 positions of C) to B. These successive positions will therefore 

 be determined precisely as C is determined, and will be on the 

 Perimeter of an ellipse of a variable transverse axis and at its 

 point of contact with the horizontal tangent. 



By means of these considerations, the proposed problem is re- 

 duced to 



Finding the locus of all the points of contact between an el- 

 lipse and a straight line, the transverse axis of the ellipse being 

 variable; whilst its inclination to the straight line is constant: 

 the distance between the foci being also constant. 



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