238 Prof. Morton and Mr. Tobin on 'Times of Descent 



the upper point as fixing the arc. Call this 6 and the 

 inclination of the chord, as before, ft. The radius of the 

 circle must be replaced in the formula? by the length of the 

 chord I which is to be regarded as constant. When this is 

 done the expression for the time, expressed as a multiple of 

 that required for vertical fall through /, is 



{K-F(<£)}/2sin*(0-£), mod. smift 

 where sin <f> — sin (/3 — \6) cosec \6. 



This is shown plotted against 6 on fig. 10. The points 

 Fiar. 10. 



10 20 30 40 



130 140 150 160 170 180 



m 90 ioo no 



8° 



marked with a ring correspond to Galileo's arc. It will be 

 seen that the direction of initial motion for the arc of 

 minimum time is inclined to the horizontal at an angle which 

 varies from about 70° (in the case of a horizontal chord) 

 to 90°. 



(5) The comparative times along the various routes which 

 have been discussed are shown in fig. 11. The initial and 

 final points are taken at a fixed distance apart and the 

 abscissa is the inclination to the horizontal of the line joining 

 them. The unit of time is that required for the vertical fall 



