408 Prof. Stokes on the Communication of Vibration 

 2U n for U. We have then 



^--L^-r+c)^» (13) 



v r F (c) 



It must be remembered that this is merely an expression ap- 

 plicable at once to all the canals, the motion in each of which 

 takes place wholly along the radius vector ; and accordingly the 

 expression is not to be differentiated with respect to 6 or co with 

 the view of applying the formula (3). 



On comparing (13) with the expression for the function <fi in 

 the actual motion at a great distance from the sphere (12), we 

 see that the two are identical, with the exception that U„ is di- 

 vided by two different constants, namely F (c) in the former case 

 and ¥ n (c) in the latter. The same will be true of the leading 

 terms (or those of the order r _1 ) in the expressions for the con- 

 densation and velocity*. Hence if the mode of vibration of the 

 sphere is such that the normal velocity of its surface is expressed 

 by a Laplace's Function of any one order, the disturbance at a 

 great distance from the sphere will vary from one direction to 

 another according to the same law as if lateral motion had been 

 prevented, the amplitude of excursion at a given distance from 

 the centre varying in both cases as the amplitude of excursion, 

 in a normal direction, of the surface of the sphere itself. The 

 only difference is that expressed by the symbolic ratio F n (c) :Y (c). 

 If we suppose F n (c) reduced to the form fi n (cos a n -\- V — 1 sina„), 

 the amplitude of vibration in the actual case will be to that in 

 the supposed case as /jl to \x ni and the phases in the two cases 

 will differ by ot Q — a n . 



If the normal velocity of the surface of the sphere be not ex- 

 pressible by a single Laplace's Function, but only by a series, finite 

 or infinite, of such functions, the disturbance at a given great 

 distance from the centre will no longer vary from one direction 

 to another according to the same law as the normal velocity of 

 the surface of the sphere, since the modulus fi n and likewise the 

 amplitude a n of the imaginary quantity F w (c) vary with the order 

 of the function. 



Let us now suppose the disturbance expressed by a Laplace's 

 Function of some one order, and seek the numerical value of the 



* Of course it would be true if the complete differential coefficients with 

 respect to r of the right-hand members of (12) and (13) were taken; but 

 then the former does not give the velocit}^ u' except as to its leading- term, 

 since (12) has been deduced from the exact espression (11) by reducing 

 fn{r) to its first term 1 ; nor again is it true, except as to terms of the order 

 r -1 , of the actual motion of the unimpeded fluid that the whole velocity is 

 in the direction of the radius vector. 



