1904] The Longitudinal Stability of Aerial Gliders. 115 



changes sign if E = 0. If stability is broken by a fall of velocity the 

 only quantities which can vanish are H and C, of these H is more 

 difficult to make positive than C, and hence it appears that the most 

 likely way for a glider to overturn in general is by commencing with 

 a series of oscillations of increasing amplitude. This again agrees with 

 experiments. 



9. Conclusions. 



1. For a glider or other body moving in a vertical plane in a 

 resisting medium of any kind whatever, the small oscillations about a 

 state of uniform rectilinear motion are determined by an equation of 

 the fourth degree, so that the conditions for stable steady motion are 

 those obtained by Routh. 



2. The coefficients in the period equation involve, in addition to the 

 ordinary dynamical constants, nine quantities X M . . . G#, which, when 

 referred to rectangular axes fixed in the body, represent the differential 

 coefficients of the forces and couple due to the aerial resistances with 

 respect to its translatory and rotatory velocity components. 



3. In the case of a system of aeroplanes these nine quantities can 

 be expressed for the separate planes in terms of /' '(a) and <£'(a), 

 where /(a) and <j>(oc) are functions determining the resultant thrust, 

 and the position of the centre of pressure when the direction of the 

 relative motion of the air makes an angle a with the plane. These 

 functions have been tabulated for certain different forms of surfaces, 

 but further data are greatly needed. 



4. The longitudinal stability of the gliders is thus seen to be capable of 

 mathematical investigation, and it is of paramount importance that the 

 present methods should be practically applied to any aerial machines 

 that may be designed or constructed before any actual glides are 

 attempted. 



5. The methods of calculation are exemplified by numerical deter- 

 minations of the criterion of stability in the cases of a single plane 

 lamina and a pair of planes one behind the other. Most of the calcula- 

 tions have been performed for an angle of gliding of 10° with the 

 horizon, and it has been necessary to assume arbitrary values for the 

 moment of inertia of the lamina. 



6. The condition that any steady linear motion may be stable in all 

 these cases assumes the form V 2 > lea, where a is a constant depending 

 on the linear dimensions of the glider, and k is a constant depending 

 on its shape, the angle of gliding and the law of aerial resistance. 



7. For a pair of narrow slats, in which the variations in the positions 

 of the centres of pressure of each are neglected, certain coefficients of 

 stability vanish. If the planes are square so that the displacements 

 of the centres of pressure are not neglected, the system is less stable 

 than a single plane of breadth equal to one of the squares. 



8. By inclining the planes at a small angle to each other the stability 



