ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 343 



it follows that there will also be variation of density. The effect of this 

 variation of density will be to cause the motion of each particle in contact 

 with the spherical surface to be curvilinear, and to be continually directed 

 to or from the varying positions of the centre*. Hence the equation (1) 

 obtained in Article (1) will be applicable to the case before us by merely 

 substituting F'cos 9 for v. The same substitution being made in equation 

 (3), the three equations (1), (2), (3) may be immediately made use of for 

 our present purpose. 



Let V = mcp(t). Then (8) becomes for this case, 



dd> Ca -Si -^1 r a J. 



-j7 = z~ e T ~ am cos ® e ' l er <P'(t)dt, 



and from equation (6), 



f A at at at 



a 2 Nap. log. P = — T e~ ' + am cos 9e'~ fe T <p'(t) dt - m 2 cos 2 6 ^j] 2 - 



/' — il 



e r (p'{t)dt = e r \// (t), and that p = 1, <p (t) = 0, and 



xj/ (t) = k, when t — 0. Hence, 



f 



= — - + kam cos 6, 



r l 



J -2il <m 2 



and « 2 Nap. log. P = am cos 9 {$ (t) - he r \-~z- cos 2 9 ^>(t)]\ (10.) 



So 



If we put 1 + er for p and neglect terms of the order of o- 2 , we shall 

 have a 2 Nap. log. p = a* a = the effective pressure on a unit of surface 

 of the sphere. The effective pressure on the whole sphere estimated in 

 the positive direction of the sphere's motion is — l^r* fa 2 a sin 6 cos 6d0, 

 taken from 9 = to 8 = tt. The negative sign is prefixed because it has 

 been already assumed that the velocity of the fluid is positive when it tends 

 from a centre, and as the central velocity in this instance is m cos 9<p(t), 

 9 must be measured from the point of the sphere which is foremost when 

 the motion is in the positive direction, so that the resultant of the pressure 

 on an annulus of breadth rd9 and radius r sin 9 is in the negative direc- 

 tion when cos 9 is positive. 



* See the proof of this assertion in the * Note ' added to this paper. 



PP2 



