288 SUPPLEMENT TO CHAP. XIV. [CHAP. TTT t 



If we put 



Ai = sin' a sin 1 + 2 cos a (cos /3 cot a a sin a + /3 sin /3), 

 Ai = 2 cos a (a sin a (9 sin /9), 

 B, = 2 (a 8 sin a COS a + 2 a COS* a), 

 B 3 = 4 a COS 1 a, 

 we have 



A At. * ~~A~ * /A 



H AI A a At /* 



^ ' p *p . ss x^ i 



1 + Bl KV 



A B A 



or, if we put A = .-', B = _-*, and H = P _- 1 , we have 



_ 1 A K 



But Ho = P -5-' is the value of H from the formula above, when the terms 

 "i 



containing Tc are disregarded. 

 We have also, by series (see Art. 20, following), 



A, = 2 (a 2 - /3 2 ) [1 - i (4 a 8 + /3 2 ) + . . .] 

 B! = -fg a* (1 - /r a* + . . .) B 2 = 4 a (1 - a* + . . .) 

 Approximately, therefore, since for A. small with respect to a, the tan- 

 gent may be taken for the arc, and hence = , or a = ^, we have 



ran 2 h 



6 a* 15 15 a* 



2 a 4 33 A 4 ' 



Hence, when rise is small compared with span, we have the approximate 

 expression 



< _ 

 M 



H = H. 6 f -'' 



15 



By means of a table calculated for H , for various values of a and /3 = 0, 

 0.2, 0.4, etc., of a, the thrust can be readily found in any case from the 

 above formula. 



We give in the following Tables the values of H,,, A and B, calculated 

 from the exact formulse. The formula for H above is thus made of easy 

 practical application, without tedious calculation, and the results given by 

 the method of Chap. XIV. may easily be checked. 



The value of V is given in Art. 3. 



