THE INTERNAL WOKK 01 THE WIND. 27 



have a = .=■ , c being a constant which can be so chosen that the angle a may have 

 any desired value. 



We will assign such a value to this constant that the angle « may always be 

 small, making it possible to neglect all powers of <> greater than unity and to 

 assume (since /(«) should disappear when <* = ()) that: 



sin a = a cos a = ! f(a) = ha 



h being a constant derived from experimental results. 



Substituting these values in the equations for the movement of the center of 

 gravity, we have : 



1 -r-. — V -;- = — <i sin /) 



V , = — g sin ft V'-7ia° 



■ m «s mg 



\ V' _.,d/J _ a SA^ 



mg 



{^--- X \ls='-> C ™P 



Replacing a by its value y r , and calling for the sake of simplification — he = K, 



we have finally : 



i 



| V./V c 



-5- = — p sin - K fl (1) 



V s ^=-flrcos/J + K (2) 



& 



[ ' ds 



1. Calculation of the velocity. — To determine the velocity in terms of the an- 

 gle nsidering ft as an independent variable, we divide each side of equation (1) 

 by the corresponding side of equation (2), which gives, after the denominators 

 have been eliminated : 



V(sr cos (i — K)rfV — V : g sin fidft = Kcd ft 



a linear differential equation in V 8 whose integral factor is 2 (<7 cos ft - K). 

 Multiplying the two sides of the preceding equation by this factor we get 



2V (g cos ft — K) hl\ T — 2V 5 (</ cos ft—K)g sin ftdft = 2Kc (gr cos ft — K) dft 



each side of which is an exact differential ; from this we get by integrating : 



V s (g cos ft — K) 2 = 2Kc (;/ sin — Kft) + C e 

 the constant C being determined by the initial conditions of the movement. Sub- 

 stituting V for V and b for ft we obtain 



V„- (<j cos b — K)» = 2Kc (g sin b — Kb) + O 



and by subtracting this equation from the preceding: 



V s (g cos ft — K) 5 = V„- (</ cos 7, — K) 5 + 2Kc (</ sin /i — g sin /> — Kft + K6) 



whence 



V= %r~\- 1 V °* (fir cos b — K)» + 2Kc (gr sin /S — g sin b — K/? + Kb) (3) 



the value of V in terms of ft. 



