PROFESSOR KELLAND ON THE THEORY OF WAVES. 125 



COR. 2. If w 2 =H and e be the elevation in the middle of the canal, e' at any 



point 



COR. 3. If y be great 



( : f: -. e . 2 nearly, 



that is the elevations increase in geometrical progression. 



65. We have, in the next place, to give the general solution of this problem, 

 which, lest we trespass too long, we shall do with as much brevity as is con- 

 sistent with intelligibility. 



In the first place, if be that function of which the partial differential co- 

 efficients with respect to the three co-ordinates, represent the velocities in their 

 direction, we have 



>-CO x>CO 



= 2/ / cos p x e my . e n z f(m, n, i)dmdn 

 Jo Jo 



subject to the condition m 2 +n s p s> =0 



,,eo /""co 



= 2/ / cospxe my e nz mf(m,n,f)dmdn 



J o *s a 



f 



tv = lf f wspxe my e nz nf(m,n,f)dmdn 



/ It I/ 



And the conditions (8) and (9) become 



2/ / cospxe my mf(m,n,f)dmdn=Q , . . (8') 



2/ / cos p x e n *mf (m, n, t}dmdn Q ,'9'j 



Jo Jo 



Also equation (10) gives at the surface, 

 i 



mcospxe" 1 * z e n *f(m, n,f)dmdn 



The equations (8') and (9') will be satisfied by supposing that n and m admit 

 each of equal values with opposite signs. This therefore gives us the following 

 values of u, v, it. 



*P sin P* (e my + e-^y) (e nf + e~ n ^f (m, n, f) dmdn 



VOL. XV. PART I. 



