1890.] Mr Larmor, Of Electrification on Ripples. 69 



April 28, 1890. 

 Mr J. W. Clark, President, in the Chair. 



The following Communications were made to the Society : 



(1) On the series in tukich the exponents of the powers are the 

 pentagonal numbers. By Dr Glaisher. 



The principal result was that the square of the series 



l + ? + r/ + g 5 + r/ + 9 12 + c/ 5 + 2 22 + r/ 6 + &c. 



(which is known to be equal to the product 



(1 + q) (1 + cf) (1 - cf) (1 + f) (1 + cf) (1 - cf) (1 + cf). . .) 



is the series 



1 + E (13) q + E (25) cf + E (37) cf + E (49) cf + &c. 



where E (n) denotes the excess of the number of divisors of n 

 which = 1, mod 4, over the number which = 3, mod 4. 



(2) The Influence of Electrification on Ripples. By J. Larmor, 

 M.A., St John's College. 



The relation between the period and the wave length of ripples 

 on the surface of a liquid must obviously be sensibly affected by an 

 electric charge communicated to the surface. 



To investigate the amount of the effect, let us take the origin 

 of coordinates on the surface, the axis of y vertically downwards, 

 and the axis of x along the direction of propagation of the ripples. 



A suitable form for the electric j)otential V above the liquid is 



V = - Ay + ACe~ my cos mx. 



The equation of the surface (V= 0) is then 



y = G cos nix, ' 



and the surface density a of the electric charge is equal to A^ir. 



The velocity potential of the wave motion is a function <j) 

 which satisfies 



dx*^dif~ ' 

 and gives at the surface (y = 0), 



deb dO 



-v- = -j- cos nix. 



ay at 



G— 2 



