THE INTERNAL WORK OF THE WIND. 29 



Finally, when K<g, the integral 



C d % = __L ^g / I I - K' sin ft - K cos /J + <A 



We Lave then, in this case : 



2gy = - V + 2 ~== Log f ^Fg rin /»- Kco. /» + y\ + Ce 

 >0 — K \ g cos /* — K / 



the constant C coining from the equation : 



= _v » + 2- -t-„/V? iK'sinft- 



Substituting; this value for C we obtain 



Log /lV-K'rinb-K co S h + fl ,\ Ce 

 \ gr cos 6 — K / 



'& 



3 Kc Loo- 1 ^ — K ' sin ^ — K cos Z 3 ■ 

 i/g' — K 2 ° I ^y _ K s sin b _ k cos b 



2gy = V,» - V + 2 ^ -^ Log ( 1^ ~ K' s,n /?- K cos /J + g x £ cos 6-K , ( 



g cos /i — K * 



One of these equations (4), (5), or (6), will give the value of y in terms 

 of /?, but in each numerical example we must choose the equation which corre- 

 sponds to the ratio of K to g in that example. 



It is well to note that the formula (6) which gives y when K< g can be put 

 into a form much more convenient for numerical calculation. This is effected by 

 calling — = cos 0, as K < g in this case ; we get in this way : 



\ 1 = sin <p 



\ a- 



whence 



V'-l 



sin ft — — cos ft -\- \ = sin cp sin ft — cos cp cos ft-\- I 



= 1 — COS (cp -\- ft) 



~ ■ . V+fl 



and 



^ ^ cp -\~ ft • cp — ft 



— — cos ft = cos cp — cos fi = — 2 sin ., sin ' 



Substituting these values in (6) and reducing, the equation becomes : 



(. <p + /s . ®— /» N 

 sin ' sin - x ~: — 

 2 2 



which can be written, since , = cos ^ = cotang <p-. 



y g' — K" sin cp 



(cp + ft <p — ft cp — b a> -f- l\ 



Log sin — g — Log sin — g— + Log sin ., — — Log sin g~ ) 



In this form, equation (6) is eminently well suited for numerical calculation ; 



