810 Prof. A. Taber Jones on the Motion of a 



if the motion of the bob on the first parabola is toward 

 the right, then the first parabolic arc starts at the point 

 ( — rsina, +r cos a), and ends, as Professor Morton shows, 

 at the point ( + rsin3a, +rcos3a). If we take the origin 

 at the initial point of the sth parabolic arc, the equation of 

 the 5th parabola may be written 



y x . M x 2 



- : — ~ tan a 



AJi* COS 2 a< 



©•■ ■ ■ ■ « 



where a, stands for the angle of elevation at the initial point 

 of the parabolic arc, and h s for the distance from this initial 

 point up to the level of no velocity. For s = l we have ui = ot 

 and h 1 jr = ^ cos a. For s — 2 we find 



fi4 cos 6 a - 1 12 cos* a + 48 cos 2 a - 3 



tan a 2 — - — g — j ToTJ a Tv ■ tan u • (3) 



64 cos b a — 14-fc cos 4 a + 96 cos 2 a— 17 v J 



and - 2 = i C os«[9-8cos 2 a]. 



r 



For 5>2 the expressions for a 6 , and probably for h s , become 

 very complicated, and I have not attempted to "use them. 



To find the path which corresponds to any given a, I pro- 

 ceed as follows. Alter using (1), (2), and (3) to find the 

 equations of the first and second parabolas, I determine the 

 co-ordinates of the end of the second parabolic arc by a process 

 of successive approximation. Knowing these co-ordinates 

 and the equation of the second parabola, I find the angle 

 of incidence of the second parabola upon the circle, take the 

 angle of reflexion as equal to the angle of incidence, and so 

 find « 3 . I can then use (1) and (2) to find the equation of 

 the third parabola. By this method the path which corre- 

 sponds to any given a may be found to any desired degree of 

 accuracy and for as many parabolas of the series as may be 

 desired. The general manner in which the path changes 

 with changing a may be traced to the beginning of the 

 fourth parabola from the curves in fig. 1. 



Special cases. — In the cases which are especially interesting 

 the paths repeat, and so form figures that are described 

 periodically and may be thought of as somewhat ana- 

 logous to the well-known Lissajous curves. These cases fall 

 into three classes : — 



(1) Cases in which the end of some one of the parabolic 



arcs is tangent to the circle, 



(2) Cases in which the end of some one of the parabolic 



arcs is perpendicular to the circle, 



(3) Cases in which some one of the parabolic arcs de- 



generates into a vertical line. 

 From the first class are to be excluded those cases in 



