28 THE INTKKXAI. WOIIK OF dlE WIND. 



2. Calculation of tlie ordinaU y in terms of the angle ft— Equation (1) can 

 lie written: 



Kc 

 V./V: - g sin fids — — ds 



But sin fids = dy, and from equation (2), * - c ~* P _ K ■ substituting these 



the preceding equation becomes: 



VrfV = - gdy + - ^J4 : K 



whence by integration : 



C 



2 gy = - V s + 2KC I ~. jp+ o 



JJ J g cos p — K 



Here we have to consider three possible cases: where K is greater than, equal 



to, or less than g. 



C <ifi _ l • / K cos /? - g \ 



When K > y, J jj cos p - k~ vk^? arc 81D {,, ,,,., p _ K j 



In this case : 



Ke /K cos (i - g\ 

 Sail V s 4- 2 — , arc sin I — I + ^ 



agy — f * ^ Ki _ g , \g cos /j _ k/ 



Since, considering the starting point of the aeroplane as the origin, the initial 

 conditions' are: y = 0^Y=V , fi = b, the constant C is obtained from the equation : 



Kc 



= _V ' + 2-^j==arc 1 



/ Kcosb-gA 



We have then, substituting this value for C : 



Kc r • /Kcos/J — g\ „;„/K cos ft — g\ ~\ m 



n n „ — \r i y; i . g arc sin I £ I — arc sinl — 5 = ^l (V 



~<W- V <, v -r- - ^gt_gt[_ \g cos /S — KJ \gcoab — KJJ 



an equation which gives the value of the ordinate // in terms of the angle fi, the 

 velocity V being already known. 

 When K = ff, 



r dp -l c dp -l r dp _i £ 



J ,, ,•,* p - k = ., J i - .-,« j8 - g J 2 sin" | ~ y s a 

 and therefore : 



~<l!l = — yj + 2c cotang £ + Ce 



From initial conditions : 



= — V,.- f 2c cotang ^ + O 



whence by substitution : 



~9!l = V « — V s + 2c( cotang ^ - cotang ^- J ( 5 ) 



