Till: INTERNAL WORK OF THE WIND. 



it is only necessary to be careful to change the Naperian logarithms of the formula 

 into common logarithms. 



:;. Calculation of th abscissa x in terms of the angle ft. — Equation (2) can 

 lie written 



K — g cos ft 



and as ds -- r ''* ., we get after substituting and integrating: 



'ft 



X 



V 8 cos ftdft 



K ;/ cos ji 



Such is the expression for x in terms of ft. We assume, however, that Y' 

 is to be replaced by its value in terms of ft derived from equation (3). The com- 

 plete calculation of this integral is probably possible ; it would be a very long 

 operation in any case, however, and as the unknown x is of little interest in the 

 result which we are workiug for, it is better to de satisfied with an approximate 

 value. This can easily be obtained, as the equation enables us to calculate the 

 velocity V corresponding to any value of angle ft and consequently to determine 

 the curve : 



V s cos ft 



z = - 



K — g cos ft 



The area of this curve relative to the axis of ft is evidently X, since 



rft 



x = zd I'j 

 J h 



4. Calculation <>/' /In ti/m i in terms of ft. — Equation (2) can be also written 



V 'Jfi = K - g cos ft 



whence : 



>ft 



\dfi 



= f 



K — i/ cos ft 



The same remark that applies to x also applies to t, which can be obtained 



more rapidly and with as close an approximation as desired by calculating the area 



of the curve : 



v 



z ~ K — g cos ft 



M MI. I : It A I. EXAMPLE. 



The purpose of this example is to show by means of given numerical quan 

 tities that an aeroplane can both rise and make progress against the wind, provided 

 that the velocity of the wind is variable. 



