Roi/al Society. 13d 



point is moving in such a manner, as results either from a motion of 

 propagation in a single direction, or two simultaneous motions of 

 propagation in opposite directions. The velocity of the propagation 

 is determined, and is, for air, the quantity commonly obtained by 

 theory for the velocity of sound. Again, it is shown that the forms 

 of the functions are not entirely arbitrary, but limited by the nature 

 of the question to a certain species, the primary form of which cor- 

 responds to the curve that occurs in Newton's reasoning, and by 

 writers on the theory of vibrating chords called the Taylorean Curve. 

 As any number of these curves will simultaneously satisfy the par- 

 tial differential equations, it is inferred that the vibrations they in- 

 dicate, may co-exist. If any portions of these curves, or of the 

 curves resulting from the combination of any number of them, be 

 joined together at their extremities, and so form an irregular line, 

 every two consecutive ordinates of which differ by an insensible 

 quantity, as this line will satisfy the same differential equations, it 

 indicates a possible motion, which is consequently of that bizarre 

 and irregular kind, which Lagrange first demonstrated to be the 

 general character of the vibrations. The particular form, however, 

 of this line is given, when the particular mode of the disturbance 

 which caused the motion is given. I have endeavoured to exhibit as 

 clearly as possible, the mechanical reasons of this kind of motion. 



In the next place, the bearing of the theory on the musical 

 sounds produced in tubes, is briefly considered, and particular at- 

 tention is paid to the mode in which the air vibrates in a tube open 

 at both ends, because on this point, the view I have taken, leads to 

 an inference which is at variance with the received theory. 



The equation which gives the motion in space of two dimensions 

 is integrated approximately, and the approximation is shown to be 

 such, that the integral will apply with accuracy to almost all cases 

 that can occur. Euler's integral of the equation that applies to 

 the motion in space of three dimensions, which has ever since his 

 time been considered to be particular, is here shown to be the 

 proper general solution, and adequate to solve all the cases of small 

 motions. This view of it is justified by its application to some 

 problems of interest, particularly to oblique reflections, and the 

 problem of resonances. In conclusion, 1 have stated as a result 

 of the whole preceding investigation, the manner in which analysis 

 points out the laws of any phaenomena, the theoretical inquiry into 

 which conducts to the solution of a partial differential equation. 



XXI. Proceedings of Learned Societies. 



UOYAL SOCIETY. 

 Jan.28.— T^HE following papers were read: — Experiments on 

 A the Influence of the Aurora Borealis on the Magnetic 

 Needle; extracted from Letters from the Rev. James Farquhar- 

 Bon to Captain Sabine. — On the Production of Regular Double Re- 

 fraction in the Molecules of Bodies by simple Pressure ; with Obser- 

 vations on the Origin of the doubly refracting Structure. By David 

 Brewster, L.L.D. I'.R.S. Lond. & Ed. 



T 2 Geoi-o- 



