520 Sir William Thomson on Stationary 



lines agree respectively with the assumed forms (1) and (2); 

 we clearly have 



7n(K-K0=mHU (9), 



and 



w(K6™ D -K / 6-» D )=:w.0U .... (10); 

 whence 



/_TT g-Hf" ( ' 



K = 

 K 



(11). 



Now at the free surface the pressure is constant, and hence, 

 by (5), we have 



— #?/ + U(w — U)= constant . . . . (12): 



from which, by (2), (7), and (11), we find 



n - TT 2.0(6 mD + €- m D )-2H 



whence 



,g= *3 . . (13), 



mU 2V 



which is the solution of our problem, for the case of the 

 bottom a simple harmonic curve. 



Suppose now the equation of the bottom to be 



h— («; cos m^ + /c 2 cos 2m^7 + /t 3 cos 3mx + &c. )mA/7T . (14); 



the equation of the surface, found by superposition of solu- 

 tions given by (13), allowable because the motion deviates 

 infinitely little from horizontal uniform motion throughout 

 the water, is 



y-D=p = 2 i=sl '- (15). 



e iwD i e -imD " ( e im T> £ -tmD\ 



im\J 2 V J 



To interpret the equation (14) by which the bottom is 

 defined, remark that, by the well-known summation of its 

 second member, it is equivalent to 



, \mhliT . (1 — k 2 ) , . mAlir ,ic(co$mx — k) ,^ n . 



h=zr — 7T- 1 — ^o — imA/7r= — =-* — ~ — ■ — 5— (lo). 



1 — 2/ccosmx + fc * ' 1 — 2k cos mos + k. v j 



The series (14) is convergent for all values of k less than 

 unity. According to the method of Fourier, Cauchy, and 

 Poisson, the extreme case of k infinitely little less than unity 

 will be made the foundation of our practical solutions. By 



