524 REPORT— 1902. 



is 



J 2" 

 / (s + i.v cos u + ii/ sin u, u) du 

 a 



where/ deuotes an arbitrary function of tlie two arguments s + i.v cos u + iy sin u 

 and ti. 



2. The particular solution of Laplace's equation in terms of Legendre functions, 

 namely, 



r" P™ (cos 6) cos m(f) ; 



where r, 6, cf) are polar coordinates corresponding to the rectangular coordinates, 

 .1', y, z, 18 a multiple of 



r 



I (z + ix cos u + ii/ sin m)" cos jw2< efw 

 Jo 



3. The particular solution of Laplace's equation in terms of Bessel functions, 

 namely, 



e''' J„j (A-p) cos m(j) ; 



where z, p, <f) are cylindrical coordinates, is a multiple of 



r 



g I (r + iv COS « + ,;/ sm u) ^gg ,„„ ^jj 



Jo 



4. The general solution of the general partial ditierential equation of wave- 

 motions 



is 



r-^r^ . . . t 



y = I I /(i' sin M cos v + v sin ?< sin v + z cos ?< + y, u, v) du dv, 



Jo Jo ' '" 



where / denotes an arbitrary function of the three arguments 



X sin ti cos V + y sin z< sin v + z cos m + - , m, and v, 

 o. The general solution of the equation 



as found in 4 can he expressed as a sum of particular solutions, each of the type 

 F(X, u, v) cos ] X(.r sin u cos v + y sin u sin y + s cos ;< + ) 



and each of these particular solutions can be interpreted physically as a simple 

 uniform plane wave.' 



3, The Longitudinal Stability of Aerial Gliders. 

 By Professor G. H. Bryan, Sc.D., F.R.S. 



One of the most difficult questions connected witli the problem of aerial naviga- 

 tion is the longitudinal stability of a machine supported on aero-planes and 

 aero-curves. In this paper the author investigates the general conditions of 

 stability for a gliding machine of any form, moving in a medium whose resistance 

 follows any law whatever. The problem may be thus enunciated : A body ia 



' The work will appear in extenso in the Matherfiatuehe Annalen, 



