230 Prof. Morton and Mr. Tobin on Times of Descent 



The special points we consider on the 50° and 55° curves 

 are those at which the positive ordinate is equal to the 

 oreatest depth to which the curve sinks below the axis. 

 Call the abscissa here ft u and the corresponding point on the 

 circle B x . If ft > ft lf then, no matter what value a has (a being 

 less than ft), /(« 7) +f{ft 7) is positive. If ft is between ft x 

 and the zero of the graph, say ft , then the sum of the 

 ordinates will only be positive provided a is sufficiently near 

 the ends of the range to ft . The point fti coincides with 

 the end of the graph when 7 is about 55°' 9. We have 

 therefore the following results for the critical region 

 45°<7<60°. 



45°<7<55 0, 9. There are two critical positions Bj B 

 for the upper point. If B is between C and B l5 then 

 the two chord path is quicker for all positions of A. If B 

 lies between B t and B , then on the arc OB there are two 

 points A x A 2 such that the times along BAjO, BA 2 0, and 

 BO are all equal. If A lies between A x A 2 , BO is quicker, 

 if A lies outside A 1 A 2 it is slower, than the way Via A. 

 When B is below B , then the single-chord is always quicker. 



55°-9<7<60°. There is now only the point B . When 

 B lies above this the two-chord path is quicker when A is 

 sufficiently close to B or 0. Below B it is always slower. 



It is easy to make the modifications necessary for the case 

 ft-y which corresponds to Galileo's original proposition. 

 Evidently its validity extends further than the quadrant to 

 which he confined his proof; it holds up to 7 — 55°'9, i.e., 

 through an arc of about 11 1°8 from the lowest point of the 

 circle. 



(2) Particles move under gravity from rest at an upper 

 fixed point B to a lower point along a rectilinear broken 

 path BAO. It is required to find the locus of the inter- 

 mediate point A when the time of descent is assigned. The 

 algebraic equation of this locus can, of course, be written 

 down but its expression is complicated. We may plot the 

 required curves by another method, identical with that used 

 by Clerk Maxwell in his diagrams of equipotentials of point 

 charges. Let the difference of level between the upper and 

 lower points be li. We take as unit of time the time of vertical 

 fall through this height. If a particle slides from rest at B 

 along any straight line its position after p units of time will 

 be on a circle of diameter p 2 h having its highest point 

 at B. On the other hand, if particles approach along- 

 straight lines, having velocity due to a fall from B, the locus 

 of points from which time p is taken to reach has, with 



