30 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



through the experimental points to show the consistency of the 

 measurements and calculations. 



§11. RESISTANCE DERIVATIVES, LONGITUDINAL 

 The longitudinal oscillations of the aeroplane are given by three 



equations of motion of §9, in which certain " resistance derivatives " 



are required. 



The quantity X« is the rate of change of X with change of forward 



speed u. Since X varies as the square of the speed, Xq = CU~ where 



C is some constant. 



Then ^J^, =2CU= ^ =X« and Z„= ^^« = -^f , so that these 



ou u u u 



coefficients are readily calculated. 



The derivatives X^, Zw, Mw represent the efifect of a vertical com- 

 ponent of velocity zv. The vertical component of velocity w acts with 

 the horizontal velocity U to cause the resultant wind to have an 

 inclination to the horizontal 



A^ = tan-'^-^57.3g, 



when A^ is a small angle measured in degrees. 

 Hence 



Z - ^ 



"^ u ^t) 

 The method practically substitutes the slopes , . , -„- of 



the tangents to curves of X, Z, M, 2XO — 0, for the actual curves. Wc 

 have assumed A^ small. If a curve be nearly a straight line, we may 

 substitute the tangent for the curve without great error. Thus it 

 may not always be necessary to assume A^ very small. In fact, a 

 range of from 5° to 8° is tolerable. 



Since we assume M^ — O, the balance should be undisturbed by 

 change of forward speed. Therefore, Mu, = o in all cases. 



Note that a positive value of M^ corresponds to a curve of pitching 

 moments giving statical stability or a righting moment. If Mio is 

 positive it does not necessarily follow that the aeroplane will be 

 dynamically stable, but if Mw is negative, instability is of course 

 certain. A'„ should be negative to indicate increased resistance for 

 increase of forward speed —u. For stability, Z^ should be large and 

 negative, indicating increase lift for larger angles of incidence and 

 vice versa. At stalling angles, Z^ tends to approach zero. 



