126 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



(m,n, /) dmdn 



The equation w,g+ J- =0 gives for the surface, 



which is satisfied by making 



e n z, _ n Zl 



d*f(m,n, t) +gn - 1 -=0 ; z, being the value of z for the surface. The value 

 **~* r 



off(m, n, t) deduced from this equation is 

 f(m,n, *) = Acos\/N . * + Bsin\/N t; where 



**,_**, 



Ne e 

 = a n 



n * 



Now equation (10') is reduced by multiplying up the denominator, and bring- 

 ing both terms to the same side. It becomes by this process 



f> fa> 



21 j r / m v* z i 'ft V- z,\ / 



/ / cospx{m(e e " ') (e 



+J \J 



'ft V- z,\ / n z. n z \ 



" ' ' ' 



, f) dmdn = 0. 



This equation is satisfied, if we can assign such relations between m and n 

 that 



m ( e * ', _ e - m * */) ( e *> + e ~ n *) -n-Vz, (e m ^^ + e~ m ^ *<) (e nz >- e~ n z ') = 0, 



which we evidently can do, since one term is positive, and the other negative. 

 Also, from the value off (in t n l t) it is evident that 



n e n *>-e-*' 



, + e n *, 



We can approximate to the value of this expression just in the same way as 

 in the previous process ; thus the first equation gives nP-^z,ri* z t -\'z,= ; 



and the second c 2 = 



Combining them, c 2 



m- + n* 



ffz, _ gz,- 

 -. + 1 



the same result as that given by our approximation in the previous solution. 



It is necessary to remark that m and n may in this case admit of an infinity 

 of different values : but since for them all the above equations hold, this circum- 

 stance has no effect whatever on the value of c. The most important consequence 

 which results from this general process is, the evidence which it affords in favour 

 of the truth of our previous conclusion relative to the form which the wave as- 



