acted upon by the Undulations of an Elastic Fluid. 361 



Hence, adding this to the former determination of the pressure, 

 it will follow that the total accelerative action on the sphere is 



Y^,rn' cos q{eft + c ) + j|^r(l -*W sin q(a!t + c ). 



Hitherto the sphere has been supposed to be fixed. But the 

 object of this inquiry is to determine the motion of a sphere free 

 to obey the impulses of the fluid. To make the preceding inves- 

 tigation applicable to the solution of this problem, I adopt the 

 principle that the action of the undulations on the sphere in 

 motion is due to the relative motion of tne fluid and sphere. 

 Let x be the distance of the centre of the sphere at any time t 

 from an arbitrary origin, and be reckoned positive in the direc- 

 tion of incidence, and. et tne excess of tne velocity of the fluid be 



doc 

 m sin q(a t t — x + a) — -j-> 



According to the above principle, this quantity is the equivalent 



of m! sinq(a ! t + c Q ) in the former reasoning. The centre of the 



sphere being supposed to perform small oscillations about a mean 



position, if for x within the brackets we substitute its mean value, 



only quantities of the second order will be omitted. Hence, 



putting C for — sc + u, we have 



dx 

 m! sin q(a't -\-c )=m sin q{dt + C) — ~r-\ 



and by differentiating, 



d^x 

 m! qa' cosq (a't + c ) =mqa' cos q (a't + C) r— - 



In this equation the foregoing expression for the accelerative 



d 2 x 

 force impressed on the fixed sphere is to be substituted for — -, 



and the equation is then to be identically satisfied. These ope- 

 rations conduct to the following equalities, qco being an auxiliary 

 arc : — 



5qc(l—h) ~ ml 2Afc*cosqw 



tang»= 8(8 + 2A ^ , c =C-a>, -= 8 + aAg , • 



If now by means of these relations we eliminate m' and c from 



d 2 x 

 the expression for -jp^, and neglect terms involving the square of 



qco, this being a very small arc the result will be 



d 2 x 3a'q . .. nN 5gW(3 + A/c 2 ) ,. ,. ,' n 



^ = 3T2A^ mC ° Sg(a< + C)+ 4(3 + 3A«y (l-^sm g (^ + C). 



It is here to be remarked that since the factor 1 — h has relation, 



