ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 351 



ADDITIONAL NOTE. 



The theoretical resistance to the motion of a ball-pendulum in the 

 air, obtained in the foregoing Essay, differs from that found by Poisson 

 in Vol. xi. of the Memoirs of the Paris Academy of Sciences and in the 

 Connaissance des Terns for 1834, and by M. Plana in a Memoir on the 

 Motion of a Pendulum in a Resisting Medium, published at Turin in 1835. 

 I propose therefore to add here as distinct a statement as possible of the 

 reason of this difference. 



According to the solution of these two eminent mathematicians, the 

 motion of the fluid in contact with the oscillating sphere, is partly along 

 its surface and partly directed to or from the centre : on the contrary, 

 in the solution I have given, the motion at each instant is wholly di- 

 rected to or from the centre. The following reasoning appears to prove 

 the correctness of the latter view. 



It is well known that the equation udx + vdy + wdz = 0, is the 

 differential equation of a surface which cuts at right angles the directions 

 of the motions of the particles through which it passes, if the left hand 

 side of the equation be integrable per se. And if it be integrable after 

 being multiplied by a factor N, the equation N(udx + vdy + wd%) = 0, 

 is equally the differential equation of such a surface, but more general 

 in its application. Let therefore 



N (udx + vdy + wdz) — d. (p (x, y, z, t). 



Then integrating, and supposing the arbitrary function of the time to 

 be included in cp, we shall have (x, y, z, t) — 0. The surfaces of which 

 this is the general equation, for the sake of shortness I shall call surfaces 

 of displacement. If the time t changes to t + dt, the co-ordinates x, y, z, 

 of each particle at a surface of displacement change to x + udt, y + vdt, 

 z + wdt, and are ultimately the co-ordinates of the surface of displace- 

 ment in a new position indefinitely near the former. Hence 



<p (x + udt, y + vdt, z + wdt, t + dt) = 0, 



Q Q2 



