acted upon by the Undulations of an Elastic Fluid. 357 



versely, if the cubical portion of fluid be stationary and the sphere 

 oscillate, the fluid which passes the plane through the sphere's 

 centre must be just as much as the sphere displaces; and this, 

 again, results from the usual analytical treatment of this pro- 

 blem. But from these considerations it necessarily follows that 

 the condition of the rigid boundary is involved in that mode of 

 treatment. By reasoning contained in this and previous com- 

 munications, I have pointed out that this condition is implicitly 

 introduced when udx -f vdy + wdz is assumed to be an exact differ- 

 ential for this instance of arbitrary disturbance. The two pro- 

 blems here considered admit of that assumption, and are on that 

 account correlative : but the problem of the sphere oscillating in 

 unlimited fluid is distinct from these, not being a case for which 

 udx + vdy + wdz is integrable without a factor, and requiring on 

 that account particular treatment, which 1 have already frequently 

 discussed and need not further advert to here. 



Proceeding now to trace the consequences of supposing that 

 in the equation (77) P is equal to </> 2 sin 6 cos 6, we have first the 

 following results obtained by the process indicated in Part II., 

 viz. 



// + F f' + F /" + F'\cos 2 

 a=a - VT* W + -~3r-) —' 



W 



U. /8(/i-:Fi) 3(/-F) 4(/'-F ) /"-F'Wfl 

 4 r 3 t- 3r 2 3r / 2 ' 



As this integration is independent of the previous one, /and F 

 may be supposed to have new values. Also for the same reason 

 U = where r = c, without respect to the former value of U. 

 Since it may be presumed from the former integration that two 

 sets of values of /and F will be required to satisfy the given con- 

 ditions, let us suppose that 



/ = m 3 sin q{r—aH + c 3 ) + m' 3 sin q(r — a f t + c' 3 ), 



F = m 4 sin q (r + alt + c 4 ) -f- m\ sin q (r + aJt -f c' 4 ) . 



On substituting these functions in the expression for U, it will 

 be found that the condition, U = where r = c } is satisfied if 

 m 4 = — m b , c 4 = c 3 , m\=m l 3i c' 4 = c' 3 , and if c 3 and c 3 be deter- 

 mined by the following equations, 



1 9 — 4q 2 c 2 9 — q 9 -c' 2 



tan q{c + c 3 ) = - - • 9 _ gV > tan q{c + c' 3 ) = qc . 9 _ 4gV , - 



Phil. Mag. S. 4. Vol. 31. No. 210. May 1866. 2 B 



