398 Sir William Thomson on the General Integration 

 where «M = 1 +/*«„/*+ if«^+ Hf^-a^+kc., 



It remains to determine the constants of integration so as to 

 fulfil prescribed conditions rendering the problem determinate. 

 This we shall actually do for two typical cases : — first, the sea 

 bounded north and south by two vertical cliffs ; secondly, by two 

 sloping beaches with gradual deepening from each to a single 

 maximum depth along an intermediate* parallel of latitude. 



12. Eirst, let the ocean be a belt of water between vertical 

 cliffs in two given latitudes, either both in the same hemisphere 

 or one north and the other south. The conditions of this case 

 are that there is no north and south motion of the water at either 

 of the bounding parallels of latitude ; and they are to be fulfilled 

 [§4,(13)] by putting 



t^ ' ' ' ^ 



for each of the terminal values of /jl (that is to say, the sines of 

 the bounding latitudes). If each of these is less in absolute 

 value than the least root of 7 = 0, each of the series in (50) and 

 (51) is convergent through the whole range of values of /n corre- 

 sponding to the supposed ocean. 



Calling, then, ///, //," the sines of the two bounding latitudes 

 (to be reckoned negative for south latitude if either or both be 

 south), we have, by using (50), in (52), 



and )■. . . (53) 



^').C+i9M.K HM.l=0, 

 which give 



K = 



/8(^»).«0*')-/8(A')-«0» B ) 

 _ a{fj').\(^)- a ^').\^'') T 



(54) 



With these values for C and K , (50) and (51) give 



j^ 2 ji> an ^ $ for every value of fi through the range of the 



supposed ocean ; and then the following formulae [which it is 

 convenient to recall from (13), (7), (3), (10), (38), and (48) 

 above] give h the height of the free surface, f the southward 

 displacement, and t) s/(\— fi 2 ) the eastward displacement of the 

 water at time /, latitude sin" 1 /^, and longitude yjr : 



