EXCESS SKMUVXIX'L rOK llOf.E TRW'Sl'ORT 41.? 



the bouiulary conditions of the problem. This is most eonvenieiUly done by 

 introducing a family of curves as suggested by the analogy with Fig. 2. The 

 analogy suggests that we should try to find curves in the v, ^ plane (the full 

 curves of Fig. 4) such that a point can move along any one of them with 

 velocity components 



The ecfuation of any such curve is 



dv 



= -*//? 



{(.. .c) = /; *^ (29) 



where j^o , the intercept of the curve on the j^-axis, is taken as a parameter 

 distinguishing the curve in question from others of the family. A point which 

 starts at height vo on the f-axis at time 5 = will reach height v at time 



s{v,v,) = f^' ~. (30) 



Thus, after time 5, the locus of all ])oints which start at all the various 

 heights j/Q will be the curve obtained by eliminating i/q between (29) and 

 (vSO) (shown dotted in Fig. 4). That this curve is, in fact, of the form (28) 

 and therefore a solution of the differential equation is easily seen by writing 

 (29) and (30) in terms of integrals takeii from some arl)ilrary but fixed 

 lower limit: 



^ dv , f^^dv 



!(..)= -/*^ + / 

 .(..)= -/'1 + / 



As V(, is varied both the integrals with upper limit vi, will vary, and either 

 can be expressed as a function of the other: 



rf=/(/-3 



whence 



which is identical with (28). 



