828 TRANSACTIONS OF SECTION 0. 



reaction varies approxiinately as the square of the velocity, atid as the square of the 

 linear dime7ision. 



The theoretical justification of this law is that the viscosity and elasticity 

 of the fluid as quantities of which the resistance is a function, are without 

 sensible influence. This is approximately true, first, concerning viscosity, so long 

 as the product of the linear size and velocity of the body is greater than a certain 

 minimum value ; and secondly, as to elasticity, provided that the velocity does not 

 approach too nearly the velocity of sound. In aerial flight as practised by birds or 

 man neitlier of these limitations applies, so that the V-square law may be accepted 

 as applicable. There is some correction required in regard to skin-friction ; in 

 resistance of this kind it would appear that the index is somewhat less than 2, 

 though how much at present we do not know. 



In the relations of the pressure reaction ouj^lanes at small anyles there is a law 

 of great utility to te deduced from experiment, namely, 



(2) For a given velocity the pressure varies directly as the angle. 



This law only applies to the small angle, i.e., to one which, expressed in 

 radians, the angle, its sine, and its tangent, are sensibly equal to one another. 

 Tiie law further does not apply to planes in apteroid aspect, in the extreme 

 rase of which the sine-square law of Newton can be shown to applj' ; neither 

 of these limitations seriously affects the validity of the law in relation to aerial 

 flight. 



A theory founded on the hypothesis of constant sweep — that is, upon the assump- 

 tion of a layer of air of defined thickness uniformly handled by the aeroplane — 

 gives results in agreement with these two laws, but with a defect in the constant 

 by which the quantities are related. This defect is accounted for and the theory 

 is rectified by taking into account the cyclic component in the periptery; by this 

 extension of the initial hypothesis complete harmony is established between theory 

 and experiment. 



The two laws so far established result in the fact discovered by Wenham and 

 rediscovered by Langley, that neglecting skin-friction and other direct resistance 

 the power expenditure decreases when the velocity is increased, and the law of 

 frictionless flight is established. 



(3) Neglecting sldn-finctio7i and other direct resistance, the h.p. varies inversely 

 as V ; or the resistance to flight varies inversely as V^. 



The modifying influence of skin-friction and of other resistances varying 

 directly as V^ results in the following laws, the proofs of which are given in the 

 author's ' Aerial Flight,' vol. i.. Aerodynamics, 



(4) The total resistance to flight is least when the resistance due to aerodynamic 



support ( oc y, ) is equal to the direct resistance (oc X') ; hence this is the condi- 

 tion of greatest range on given fuel-supply . 



(•5) The flight ivill be sustained for the longest time on a given supply of 

 energy when the resistance due to aerodynamic support is three times the direct 

 resistance, 



(0) The speed of greatest range is -f,.T~ ( = 1'315) times the speed of least 

 poiver. "^ 



(7) Neglecting ' body resistance ' for aerodromes or aerodones designed for 

 least resistance, the resistance is indepetident of the velocity of flight. In other 

 words, the gliding angle is constant, or the power varies directly as the velocity of 

 fliyht. . 



Cor. When body resistance is present the total resistance consists of two 

 part,s. One of which varies as the velocity squared and the other of lohich is 

 constant. 



(8) That, consequent on laws 4 and 5, there are best values of angle (the 

 angle of an aeroplane to the line of flight, or the angle of trail of a pterygoid 

 aerofoil), and of the P/V^ relation, that correspond to the condition of least 

 resistance. These, tabulated from theory, are found to be in harmony with 

 experience. 



