66 Prof. Challis's second Reply to Mr. Airy 



ance to an oscillating sphere than to Poisson's, I select the 

 latter for an example, as affording by the verification of the 

 equations the most convincing proof of the possibility of the 

 motion under discussion. 



Poisson first transforms the equation (A.) into one of which 

 the coordinates are polar, the centre of the sphere being pole, 

 and takes into account the motion of the sphere. (See Addi- 

 tions to the Connaissance des Terns for 1834, p. 40. The 

 equation under the form required for my present purpose is 

 given in my communication to the June Number.) As the 

 coordinates of the centre of the sphere do not appear in the 

 transformed equation, the sphere may be regarded as sta- 

 tionary. The integral obtained by Poisson in art. (8.) of the 

 memoir above cited, is, 



r being the distance of a point in the fluid from the centre 

 of the sphere, and 6 the angle which r makes with the line 

 in which the centre moves. Hence, putting R for the quan- 

 tity in brackets, 



d<p dR Q , d<p R . n 



—r 2 - = —, — cos ; and — rs = sin 0. 



dr d r r d0 r 



These are respectively the components of the velocity in the 



direction of r and in a direction perpendicular to r. If the 



centre of the sphere be the origin of rectangular coordinates, 



and the axis of * be taken in the direction of the sphere's 



motion, it may readily be shown that 



__ / dR R\ 



\ r 2 dr r 3 / 



( dR R\ 



v =\Fd7--?r) ss y 



/ dR R\ . ; R 



TO = ( -5-7- 5" ) « 9 + 



\r*dr r 5 / r 



p= -kM = -k dR * 



dt ' dt ' r' 



It is now an easy matter to show that the preceding values of 

 «, v, w, and P, satisfy the equations (1.), (2.), (3.), (4.), (5.), 

 (6.). For example, from the values of P and w, 



* P = k f d * R ** d * R *° d * R 



dz dt 



f d 3 R z^_ d*R z*_ d*R 1T\ 

 \dr dt*' r* dP ' r 3 + dt* ' rj 



, d*w . d*P , d*>w , . . . 



= " * • w and dtTz = ~ k - -zw> b y e q uation ( c 0. 



