1904.] The Longitudinal Stability of Aerial Gliders. 103 



Equation (4) or (4a) is the period equation for small fluctuations 

 about steady motion. 



In order that the steady motion may be stable, the roots of this 

 biquadratic must either be real and negative, or complex with their 

 real parts negative, and Routh* finds that this will be the case if the 

 six quantities 



A, B, C, D, E, and BCD - AD 2 - EB 2 , 



are all of the same sign. Since A is essentially positive the remaining 

 five quantities must all be positive. We shall denote the last quantity 

 byH. 



3. General Theorems.^ 



(1) The general transformation formulae, connecting the nine co- 

 efficients X u . . . Ge referred to any given system of axes with those 

 referred to any other system of axes, may be easily written down and 

 need not be discussed here. 



(2) The work of calculating these coefficients for a given system 

 may be reduced if the resistance is proportional to the square of the 

 velocity, for X, Y, G will then be homogeneous quadratic functions of 

 u, v, 0, and Euler's theorem of homogeneous functions gives, remember- 

 ing that 0o = and applying (2), 



uX u + vX v = 2g cos 0, 



uY lt + vY v = - 2g sin 9, 



uGc u + v& v = 0. 



(3) If V is the velocity of gliding, the coefficients mX u . . . mG# are 

 all linear functions of V for a given angle of gliding. But mg being, 

 in steady motion, equal to the vertical resistance, is proportional to V 2 . 

 Hence if m be eliminated, the values of X w . . . G^ are inversely 

 proportional to V. In this case 



A is independent of V, 

 B is of dimensions V -1 , 

 Cis of formP + QV" 2 , 

 D is of dimensions V -1 , 

 E is of dimensions V" -2 , 



and the expression H or BCD-AD 2 -EB 2 assumes the form 

 PV- 2 + QY- 4 . Thus the conditions of stability C > O and H > O 

 impose limits on the value of V 2 , the remaining conditions only 



* Routh, ' Advanced Rigid Dynamics,' p. 167. 



f This section and also the subsequent parts enclosed in [ ] have been 



rewritten October 27, 1903." 



