Theory of Long Waves. 261 



.&= (a + bz)-*\ fi(u—^/4ng(a + bz)lb . cos <f>) sin" 2 " <j> . d<j) 



+ (a + 6ar)—l / 2 ( M + ^/^{a + bz)/b . cos 0) sin- 2ra c/> . d(j>,(29b) 

 Jo 



when h is negative: if n, however, lies between +J and — £ 



either form may be taken. 



From this general solution we see immediately that if 2n 



be an odd integer, positive or negative, the solution will take 



a finite form on integrating. We shall obtain, in fact, 



3= §- z AMu- v^i- !)(<* + bz)[b) 



or 



+ F 2 {u + \/2g{2i-l)(a + bz)lb)}, . . (30) 



■+F s (M.+ v / -2^2»-lXa+^)/i)}, • • (31) 



according as n=-i—\ or n = ^ — /, z being a positive integer. 

 These cases of finite solution are specially interesting for the 

 smaller values of i, as being then more easily interpreted : it 

 may be noted that the case of the channel of uniform propa- 

 gation is obtained by taking / = 0, in (30), or 8 = 1, in (31). 

 It should be observed that, as in any real channel the cross- 

 sectional area <$>(z) must increase with 0, b and n must be 

 quantities of the same sign. 



Whatever be the form of <j>(z), the general equation (4) 

 might be treated by thus changing the independent variables 

 to u and p ; but the solution would in general be very com- 

 plex in form. This mode of solution, however, is interesting 

 theoretically from its relation to the finite solution we have 

 obtained for waves propagated in one direction only. For 

 that solution is the most general one that satisfies the con- 

 dition u=f(p), while this latter is general only when u and 

 p are independent ; thus the solutions are supplementary to 

 each other, the cases excluded from the one system are pre- 

 cisely those included under the other. 



5. Tides in Rivers and Estuaries. 



The tide in a river or long narrow estuary is due to the 

 propagation upwards from the outlet of long waves which 

 may be regarded as forced by the oceanic tide at the mouth. 

 This tide at the mouth may, for our present purpose, be 

 regarded as compounded of two simple harmonic constituents, 



