276 Lord Rayleigh on Waves. 



When -y- is considerable, (L) becomes 



-, (M) 



or, for a closer approximation, 



— (l + 2«-TT) (N) 



Formula (M) was found by Professor Guthrie to agree with 

 observation when ?i=l or 2. The periods in the two cases are 

 in the ratio 1 : <s/2, if the depth be sufficient. 



If the depth bear a constant ratio to the length, (L) or (I) 



shows that the period is directly proportional to the square 



root of the linear dimension; and the same law will obtain 



when the depth is great, whatever the absolute value may be. 



If n = l, the points of constant elevation occur when 



#= — (that is, in the middle of the length) ; and if n — 2, when 



u 



T ^T 



x= — or -j-. The maximum elevations (or depressions) are 



equal. 



These results take into account inertia and gravity only. 

 From some expressions in his paper Professor Guthrie would 

 appear to attribute the effect of shallowness in increasing the 

 period to friction. No doubt friction must act in this direc- 

 tion ; but its immediate effect is on the amplitude, and not on 

 the period. In all ordinary cases the action of insufficient 

 depth may be sufficiently accounted for by the increase of 

 the effective inertia due to the contraction of the chan- 

 nels along which the liquid flows, in the same way as the 

 pitch of an organ-pipe is lowered by an obstruction at the 

 mouth. In such vessels as those used by Professor Guthrie 

 it may be doubted whether friction and capillarity have any 

 sensible influence on the periodic time. 



The theory for the circular trough depends on the class of 

 functions named after Bessel, which are an extreme case of 

 Laplace's spherical functions. For the symmetrical vibrations 

 we have 



M=J («r), (0) 



being the radius vector ; and if R be the radius of the vessel, 

 k is a root of 



J'„(«R)=0 (P) 



If ^? = /cR, the values of x satisfying (P) are 3*832, 7*015, 

 10*174, &c, of which only the first belongs to the cases experi- 



