acted upon by the Undulations of an Elastic Fluid. 353 



curvilinear, it cannot be true when the lines of motion diverge 

 from a fixed centre, because the motion in that case is rectilinear, 

 and the velocity must consequently be a function of the distance 

 from the centre. 



The above-mentioned principle being thus shown to be inap- 

 plicable, the following process of solution has been adopted for 

 the case of undulations incident on a fixed sphere. As in this 

 case a relation exists between U and W which depends on the 

 mutual action of the parts of the fluid, and is on that account to 

 be ascertained by integration, the quantities in brackets in the 

 equation (S) are equated to zero, and from the two resulting 

 equations and the equation (e) U and W are eliminated. These 

 operations conduct to the equation (£) in Part II., by differentia- 

 ting which with respect to the equation employed in the solu- 

 tion of our problem is obtained, viz. 



d*V _4!P l/rf 2 P , dV P \ ■ 



~a!W*- dr* + ?\W*jW M °~wFd)y * [V) 



P being put for r -rtf and a! for ica, for the sake of brevity. This 



equation is to be used rather than the one from which it was 

 immediately deduced, because, while it includes the latter, it 

 admits of more extensive application, as will appear in the sequel. 

 By supposing that P = </>j sin + </> 2 sin #cos 6, it is found that 

 the equation is satisfied if </>j and c/> 2 be functions of r and t 

 determined by integrating the equations . 



2 JJ1 <l ^3 



a n dt 2 dr 2 



a n dfi dr 2 + r 3 



Now, if all the conditions of the problem can be satisfied by 

 this particular integration, it will follow that we have obtained 

 the only appropriate value of P, and that no other solution of 

 the problem is possible. It will be now assumed that the inci- 

 dent undulations are defined by the equations 



V = a! a 3 = m! sin q [alt + r cos 6 -f c ), 



and .that the direction of incidence is contrary to that in which 

 the radius vector is drawn when = 0. As in the applications 

 proposed to be made of these researches the radius of the sphere 

 will always be extremely small, it will be supposed that, while 

 the distance r Y from the centre of the sphere within which its 

 reaction on the fluid is of sensible magnitude is very large com- 

 pared to c the radius of the sphere, it is very small compared to 



