Prof. Challis on Hydro dynamics \ 263 



motion, the above expressions give, to the first approximation, 

 the velocity and condensation due to the reaction of the sphere. 

 We have thus the case of waves incident on a sphere at rest, 

 but unaccompanied by condensation, the whole mass of fluid 

 moving with the same velocity. The motion of the fluid being 



dT . 

 by hypothesis vibratory, the expression for -7- will be a periodic 



function, even if T be supposed to be identical with w 1 , and 

 consequently to include a term of the second order. Hence by 

 such action there will be no tendency to produce a permanent 

 motion of translation of the sphere. The case will, however, be 

 different if the impinging waves be accompanied by condensa- 

 tion, and to this circumstance we have now to direct our atten- 

 tion. But a preliminary remark of considerable importance 

 must first be made. 



The above expressions show that if the velocity T be constant, 

 and consequently either the sphere have a given velocity, or a 

 uniform stream impinge upon it, the condensation a vanishes, so 

 that there is no dynamical action between the sphere and the 

 fluid. At the same time the velocities U and W are such that 

 in the case of the moving sphere as much fluid passes it as it 

 displaces ; and in the case of the fixed sphere, the quantity of 

 fluid which passes the plane through its centre perpendicular to 

 the stream is the same as if the stream had not been interrupted 

 by the sphere. The circumstances will be very nearly the same 

 if T represent the velocity in waves of which the breadth and 

 maximum velocity are such that the excursion of a given par- 

 ticle much exceeds the diameter of the sphere. In fact I have 

 only recently recognized that the part of the integral of (77) now 

 under consideration applies especially to motions of this mag- 

 nitude, and that it is distinguished in that respect from the 

 other part, which applies to motions that are small compared to 

 dimensions of the sphere. This will more fully appear in the 

 sequel : at present it is to be understood that we are investi- 

 gating the dynamical action, on a small sphere, of waves of alter- 

 nate condensation and rarefaction, the excursions of whose par- 

 ticles are large compared to the sphere's diameter. 



We have, first, to obtain (j) l by integrating the equation (6). 

 The known integral of this equation is 



<£,= - • \f{r-fcat)+¥{r + fcat)} —f(r — /cat)-'E , {r + /cat), 



and for the present purpose it will be proper to retain both the 

 arbitrary functions. Also for the sake of brevity these func- 

 tions will be represented by/ and F. Hence, since qr=<fi l sin 6, 



