PROFESSOR KELLAND ON THE THEORY OF WAVES. 



then since w or F=0 when *=0; 2 e* m v . n b=0 .... (8) 

 and since v or F=Q when y=0; 2e" nz . mb=Q . . .' (9). 



'/ * d 1 1 



Also at the side of the canal, it is evident that - L =-/-"\> '(). 



m, dz 



Now v t which is the value of v at the side of the canal, may not be obtain- 

 able from the value of v given by the equations (4), (5), (6), (7), since it is pos- 

 sible that the discontinuity of the fluid may require a discontinuous function as 

 the expression for its motion in the neighbourhood of the sudden transition from 

 the fluid to the surface of the canal. Provided, however, the canal be not abrupt 

 in its curvature, and the motion be not very great in comparison with the mag- 

 nitude of the canal, it is clear that the values of v, and ?, will approach very near 

 to the values of v and rv at the surface of the fluid. In fact, we may safely argue 

 that the conditions of continuity are not more violated by putting v and w for 

 v t and m, at the surface, than they are by putting v and w themselves for the ve- 

 locities obtained from conditions belonging to the interior of the mass. With 

 these observations we shall adopt the following equation : 



==%]/ (z) at the surface, 

 or 5^N=^'(^) at the surface ..... (10). 



* (Z'-YZ) 



We have written F (z, y] for F, &c. 



Also, if we adopt only the large terms of equation 11, Art. 50, we shall have 



Tg J 



mff + -T-|- = at the surface, 



or 



... e 2 =-^. -at the surface .... (11). 

 / 



Condition (9) is satisfied by giving to m two equal values with opposite signs 

 for every value of b and n. But since m 2 + w 2 =l, the value of m is =fcVl- 2 , and 

 consequently this condition merely directs that both values must be retained. We 



1*1 i -j. f v t.f *+ /i-tf) , Jt(m ijl * v)j 



may now eliminate m, and write /= 2 o{e * 



F= 

 Condition (8) requires that F=0 when *=0. If y=v when z=0, this gives 



2 b n {e" " V l -"" + e " ^ l ~ n '} = 0. 



To confine ourselves to the most simple case, let us suppose =0 ; then we 



