under Gravity, suggested by a proposition of Galileo's. 237 

 For h>2a write h~2a cosec 2 y' 



and sin </>' = sin j3 sin 7', 



then we find arc-time = v /(«/^)sin y'F(fi) mod. sin 7', 

 chord-time = 2 s/ (a/g) tan Jcjb'. 



When the particle starts from rest at the highest point of 

 arc and chord 7 = /?, and the times become ^(a/g)K and 

 2V(a/#) respectively, where K is the complete elliptic 

 integral to modulus sin j3. These are equal when fi has the 

 value 53°'35 approximately. For smaller values of j3 the 

 time down the arc is less than that down the chord. 



Consider first a value of /3 lying in this region and let the 

 starting level be raised continuously. It is evident that 

 when the initial velocity is large the times will be nearly 

 proportional to the lengths of arc and chord, and so the arc- 

 time ultimately becomes the longer of the two. There is 

 therefore, some definite starting-level giving equal times. 

 Examination of the formulae shows that the corresponding 

 value of 7 ranges from 45° to 53°' 35 as /3 increases from 

 0° to 53°*35. It thus appears that Galileo was correct in 

 the assumption on which he founded his proof of the 

 " scholium/'' seeing he expressly limited himself to a lower 

 quadrant of the circle. 



For /3>53°*35 the ratio (arc-time)/(chord-time) has a 

 value greater than unity when the particle starts from rest. 

 Its mode of variation, as the starting-level is raised may be 

 summarized here. There is a second special value of fi given 

 by K=sec/3, /3 = 63°*9 approximately. For values of ft 

 less than this the ratio in question increases steadily as the 

 starting-level rises to infinite distance. For greater values 

 the ratio first decreases, reaches a minimum, and then 

 increases to its limiting value of (length of arc) /(length of 

 chord). For example, if /3 = 70°the ratio begins at value 

 1*252, drops to minimum of 1*236, and has value 1*239 

 when the starting-level is at the highest point of the circle 

 and approaches the limiting value 1*240. As (3 approaches 

 90° the level giving the minimum ratio recedes to infinity. 

 For motion along the semi-circle and the vertical diameter 

 (/3 = 90°) the ratio decreases continuously from jo to Jar. 



(4) The circular arc of quickest descent one would expect 

 to be determined by a compromise between the shortness of 

 path on one hand, and on the other the steepness of la 11 at 

 the upper point. The feature by which Galileo's are is 

 characterized, viz., tangent at lower point horizontal, does 

 not play any essential part in the matter. We may take the 

 inclination to the horizontal of the tangent to the circle at 



