114 



Dr. G. H. Bryan and Mr. W. E. Williams, 



[Jan, 7, 



Taking the expression 



H = BCD - AD 2 - EB 2 , 



it may be written 



CDG0 - EG<j 2 + ¥ {(X u + Y v ) CD - D 2 - 2 (X„ + Y^E} - k*(X u + Y,) 2 E. 

 H is therefore increased with increase of W if 



(X u + Y r ) (CD - 2G*E) - D 2 - 2k 2 (X„ + Y v ) E > 0. 

 At the critical velocity, if H = 0, this becomes 

 CDGe + (X u + Y„) 2 E/^ < EG*, 2 . 



Unless this condition is fulfilled, the critical velocity given by H = 

 increases with increase of Jc 2 . 



In all the numerical examples considered above, CDG# >EG# 2 and 

 E is positive, so that the critical velocity is increased by increasing h 2 . 

 Thus in Ex. (1) Ic 2 was taken equal to Ja 2 , and the critical velocity 

 obtained was N /(774«) ; if, instead, we had taken k 2 = a 2 , the critical 

 velocity would have been J (124:0a).— Jan., 1904.] 



8. Character of the Fluctuations about Steady Motion. Mode in which the 



System Overturns. 



The character of the fluctuations about steady motion depends on 

 the nature of the roots of the biquadratic (4a), the expressions for the 

 displacements 8u, 8v, 86 being evidently of the following forms : — 



For roots all real . 



For roots two real, two imaginary 

 For two pairs of imaginary roots . 



CieW -f- c 2 e K ^ + c 3 e A s' + c 4 eV. 

 CieW + c- 2 e x ^ + ye at cos (fit - e). 

 yie*!* cos (fat - €i) 



+ ytfM cos (/3 2 t - e 2 ). 



The last form indicates two different sets of undulations of different 

 lengths. Photographs of the paths of gliders taken by magnesium 

 light distinctly show these two undulations, thus confirming our 

 theory. 



An important further consequence is that a glider may perform 

 undulations decreasing in amplitude, corresponding to a pair of complex 

 roots of the biquadratic with their real part negative, but the motion 

 may be unstable through the other roots having their real part positive, 

 or one or both of them being real and positive. This indicates a real 

 danger in experimenting with gliders. 



Stability may be broken either if a real root of the equation (4a) 

 changes from negative to positive, or if the real part of a pair of 

 imaginary roots changes from negative to positive. 



The condition for the latter is that H = 0, whereas a real root 



