34 Mr. J. Proudman on the 



The value of % on OA, which gives the velocity at the 

 free surface, is 



,2a sinh (ft + i)(7r — x) sin (n+ \)x 



(9) 



and the function F, when OA is taken to be a instead of tt, is 

 given by 



^ = A"^J (^ coth(n+i)7r -- • (10) 



Remarks on the Expansions. 



4. The normal derivatives of (5) and (8) must vanish over 

 SL and A respectively. For (5) this gives us 



elMHt 1 



s tt „ =0 {n + i) 2 cosh (n + J)tt 



X {cosh (n + i)f sin (n + i)f+ sinh (a + J)f cos (n + £)f}, (11) 



for — 7r<^<7r, while for (8) it gives us an expansion which 

 is easily transformed into (11). 



Again, alternative forms can be obtained for (5) and (8), 

 and on equating them respectively to the above forms, 

 identities are obtained. Identities of this nature were noticed 

 by Sir G. Stokes *, and remarked upon by Thomson and 

 Tait f- Those mentioned by these authors were examined by 

 F. Purser J, who pointed out their connexion with Elliptic 

 Functions. 



Two additional remarks, however, seem worth making. 



The first is that the identities can be easily obtained by 

 taking a two-dimensional harmonic in algebraic Cartesian 

 form, and then finding a series of two-dimensional harmonics 

 in normal Cartesian forms (i, e. in terms of trigonometric 

 and hyperbolic functions), which satisfies the same conditions 

 at a certain boundary. 



The second is that when the expansions of conjugate 

 functions are combined to form a series of functions of 

 a complex variable, the resulting forms appear to be 

 interesting. 



For example, we can thus obtain the following expansions, 

 valid over a square whose corners are given by 



* Math, and Phys. Papers, vol. i. p. 190 (1846). 

 t Natural Philosophy, part ii. p. 249, 1883 ed. 

 X Messenger of Math. vol. xi. (1882). 



