392 MISCELLANEOUS STUDIES 



Equation (155), using y instead of (y 3), gives approximately 

 ff and 



(159) 



66' 



Substituting this value of - in equation (158) and letting 



dy 



0" = ve-^v results in the ordinary differential equation 

 dv 



i kv tt'o&jV [tt' rcos (at ej] ' f_ [f>na ( a e 2 ) 1] 



/ 2 I ] 2 



rw^vcos (a* cj (160) 



where v is a function of only. Let 



k ?/'(,&! = fc, and M z = ' 1 _ 2 

 then equation (160) can be put in the form 



)S (at ej ^ [COS (2at e 3 )-|-COSc 4 ] [ 



\ 



-f- rwj)^ cos (at x ) (161) 



where e 2 4~ e i c 



When the vertical velocity is directed upward -u f is negative. 



therefore 



k 2 =(k u? 6 1 )> | MJ &i| 



and the last term of equation (161) can be neglected in the first 

 approximation, which is 



v = e- | r 2 f e *' [cos (a* Cl ) -cos ( 2at e 3 ) -cos J r + C I 



77- f Fa cos f, -\- A\ sin e,~| . FA'., cose, a sin ]~| 

 : -^ \ I $+?- - J sin ' + [ -^+7^ Jeos a* 



r2acos C3 +A: 2 smc 3 "| 

 ~L 2(4a 2 +^) J S1 



[/> cose, 2asin.,~| cose 4 ) /ico\ 



2(tf+t.') J eos ~~2^ [ 



The arbitrary constant C is 0, since from physical considerations 

 the solution must be a periodic function of the time. Substituting 



