under Gravity, Suggested by a proposition of Galileo's. 231 



reference to as origin, the polar equation 



r = h( 2p —p 2 sin 0) , 



where 6 is the inclination to the horizontal. This locus is a 

 limacon derived from the circle of diameter/) 2 //. Let a set 

 of. circles he drawn below B and a set of limaeons round 0, 

 each curve being marked with the corresponding value of p 

 Then a series of points on the locus sought can be obtained 

 by marking intersections of circles and limaeons whose p's 

 have a constant sum. By drawing one set of curves on 

 tracing-paper and moving this sideways over the other set, 

 placed in its proper relative position, it is possible to get 

 the curves for different slopes of the line OB, keeping the 

 same value of h. 



The method fails when the line OB is horizontal, but 

 in this case the algebraic equation for the locus simplifies to 

 a manageable form. Taking the origin at the centre of the 

 line, the axis of x horizontal and y vertically downwards, 

 the equation is 



«?=q*y{itf-qHy + l*)l{l- q *y), 



where I is the length of the line and q is the time 

 expressed as a multiple of that required for vertical fall 

 through I. 



Figs. 3, 4, and 5 show the forms of the loci when the line 

 joining the terminal points is vertical, sloping, and horizontal. 

 The length of this line is made the same in each case, and 

 the numbers attached to the curves give the time as a 

 multiple of that required for vertical fall through this length. 

 The slope in fig. 4 is such that the time of sliding down the 

 inclined line is 1*5 times the unit taken. The line itself 

 forms part of the locus for this time. It will be seen that 

 the loci for longer times of descent pass through the upper 

 end of the line. We can deduce a simple expression for 

 their curvature at this point. Let B' be a near point on the 

 locus for time t' and let t be the time down the straight 

 line BO. Then 



(t r — £)=time along BB'0-time along BO. 



But in the limit the time along B'O becomes equal to that 

 along BO. Therefore [t' — t) — limit approached by the time 

 of sliding along BB' as B' approaches B. Since this has to be 

 finite the tangent at B must be horizontal and ($' — $) = time 

 down any chord of the circle of curvature = (4/3 '</)*, 



