256 Curve desaHbed by a Movable Pulley. 



y'2 x' 2 



Equation (4) then becomes n ■ + o • o = l (5) 

 1 v ' c 2 sin 2a c 2 sini« v ' 



a = 45°, gives the greatest axes of the hyperbola. 



The principles of this solution may be applied to several other 

 problems of the same kind. If it were proposed, for example, to 

 find the locus of all the points of contact of a straight line with 

 a hyperbola whose foci are fixed, and whose transverse axis varies 

 continually, whilst the tangent to the successive hyperbolas makes 

 a constant angle with the axis. 



In the same manner as in the preceding problem, we have : 



v 2 x 2 xic 2, — a 2 ) 



— — r7 = l.... b 2 =c 2 —a 2 ....m= - — .... whence we obtain 



a 2 b 2 ya 2 



the same equation (4) and consequently the same curve. 



To account for this identity of solution we must remark, that 

 the movable pulley C may be regarded as a point of the hyper- 

 bola whose foci are A and B ; and the direction of gravity, al- 

 ways bisecting the angle ACB of the lines drawn to the foci, is 

 tangent to the curve at C. 



Now this line makes a constant angle with the line of the foci ; 

 therefore the curve described by the movable pulley may be deter- 

 mined by finding the locus of the points of contact of a variable hy- 

 perbola with a right line making a constant angle with the axis. 



This result shows a new relation between these two conic sec- 

 tions which are already well known as kindred. 



A new field is now open for many problems of the same de- 

 scription, but perhaps of no very useful application. 



As for parabolas whose parameter varies continually and whose 



dy p , 



equation is, y n ~px, we have m— j-~ ~^7> the resulting lo- 



cus will be the straight line y=mnx. 



y n x " 



For ellipses and hyperbolas of higher orders — zfcr^l 



the solution will be an equilateral hyperbola of the same species. 



This when b n =a n ^c n ; and « = 45° (which makes m— 1) 



x n y n 



J 1. 



is 



c n c» 



For the curve (x 2 +y 2 y —a-y 2 +b 2 y 2 , (which is that of the 

 orthogonal projection of the centre of the ellipse on the tangent) 



we find x 2 -\-y 2 — c 



x + my 

 supposing between a, 6, and c, the same relation as in the ellipse. 



For the circle, m being independent of r, the problem is inde- 

 terminate : but it will be easily understood — that the locus is a 

 right line coinciding with the radius continually diminishing. 



The practical application of this problem may sometimes be 

 interesting : because by settling conveniently the fixed point and 



