Appendix B. 



Mr. G. E. Curtis calls my intention to the fact that the conclusion that the power 

 required to maintain the horizontal flight of an aeroplane diminishes with the increasing 

 speeds that it attains, may be deductively shown by the following analysis: 



Representing the work to be done per second by T, the resistance to horizontal motion 

 by R, and the horizontal velocity by V, we have by definition 



T= RV. 



Substituting for R its value, Wtana (see p. 65), W being the weight of the plane, we 

 have the equation 



T= VWtana, 



in which « and V are dependent variables. The curves of soaring speed (Fig. 9) enable 

 us, in the case of a few planes, to express « in terms of V, but, for any plane and without 

 actually obtaining an analytical relation between Tand «, we may determine the character 

 of the function T, i, c, whether it increases or decreases with V, in the following manner: 

 Differentiating with respect to V, we obtain 



T ( da\ 



-y — W [tan a -f- V sec 1 a -j ,.) . 



Now, since in flight a is a very small angle, tan a will be small as compared with the 

 term T' sec- a ^-jt- Hence the sign of the latter factor ~ will control the sign of ^-y 



Now, since V increases as « diminishes, -ry is negative, which makes the term V scc~ a ^ p 



negative, and therefore, in general, T is a decreasing function of V. In other words, 

 neglecting the skin friction and also any end pressure that there may be on the plane, the 

 work to lie done against resistance in the horizontal flight of an inclined plane must 

 diminish as the velocity increases. 



ir, (113) 



