1 8 Ravleigii, TJic Mechanical Principles of Flight. 



be a limit below which « cannot be reduced, if only 

 because of the high degree of instability that such an 

 adjustment must have to contend with. Another important 

 matter is the tangential force upon the plane, although 

 some distinguished experimenters have expressed the 

 opinion that it is negligible. In order to take account of 

 it, we may add to the right-hand member of (lo) a term 

 proportional to F-, but independent of a. Thus (12) 

 becomes 



H = IVU = {kS sin-a + ^1)^" . . . (17) 



(9) remaining unchanged. Eliminating /^, we find 



U' (kS S,\n-a + Jul)" 



IV~ ^-'^^^sin^a 



(18). 



We ma\- apply (18) to find for what value of sin a the 

 quantity U- attains a minimum. By the ordinar\- rules, 



^'>i"« = ,r;^ (i9)> 



and, of course, this value of sin-'« must be small, if the 

 investigation is to be applicable. If fi vanish, sin n 

 diminishes without limit. In general the miiu'mum value 

 of U- is given by 



16//'/ ,, \4 



and the corresponding value of [/-' by 



^'= ..V 



'"=?^ (-)■ 



These equations shew that the necessary work depends 

 entirely upon ;;, and that without a knowledge of this 

 element no numerical conclusions can be arrived at. 



It might be supposed that /u, so far as it depends upon 

 the aeroplane, would be proportional to ^\ but this relation 

 is more than doubtful. In any case of a practical machine 

 there must at any rate be a part of pi not proportional to S. 



