324 E. C. ANDREWS. 



The path of the particle is obtained by eliminating t from 

 the two equations 



x = u cos 6t (1) 



y = u sin 6t + i gt 2 ...{2) 



From(l) t = — - — - 

 u cos 



Substitute in (2) .*. y — x tan + \ a —^ 



tt 2 cos 2 



This gives the equation of the path of the particle. It 

 is that of a parabola. 



If u be large the equation approximates to y = x tan 9, 

 i.e., the path will be near to the straight line y = x tan 6. 

 or Ora in the figure. 



If a be small the path approximates to 



y = x tan + x 2 oo 



i.e., x — o 



i.e., to the straight line Oy. 



(After E. M. Wellisch, Emmanuel College, Cambridge.) 



Applications. — This tendency of a stream particle to 

 move in a parabolic path may be employed to ascertain the 

 nature of the valley head which is determined by sapping 

 or stream action, or by both actions combined. 



The tendency of earth particles forming the walls of a 

 vertical cut in the earth's crust is to fall down under the 

 action of weathering until the "slope of repose" is formed. 

 This is a profile convex to the sky as opposed to the profile 

 of corrasion which should be concave to the sky. Both 

 the profile of corrasion and that of repose tend to the 

 parabolic form. The reason of the respective convexity 

 and concavity with respect to the sky of the profiles of 

 corrasion and of aggradation is obvious. Although both 

 tend to the parabolic form, nevertheless the one is a build- 

 ing, the other a cutting, curve. 



