198 M. K. Zoppritz on some Hydrodynamic Problems 



omitted. Accordingly, for such points it is : — 



J&m cos mx+p sin ma __ m 2„*f*/r#-\/ >- , • «.\j* 



/r = 22— ^a +j> (^ +1 ) c J /(f)(™coswi£+/>sinwf)(/f 



^^ "wT^Tirr ^"^^ • (8) 



The first of these two members, which proceeds from the 

 initial state, vanishes if/(#) = 0, consequently if at the time 

 t = the entire mass was at rest ; but it vanishes also with any 

 value of f(x) if t is very great — that is, if the initial state lies 

 in a very remote past. In both cases the motion in the inte- 

 rior of the fluid is represented by the last member alone. The 

 regularities which result from this expression relative to the 

 motion at different depths and at different times are for this 

 problem partly the same as those pointed out by Fourier in 

 the simpler problem when the temperature of the surface is 

 given ; for they agree so far, that they depend on the integral 

 according to X, which is common to both problems. 



If <f)(t) = u\ is a given quantity independent of the time, 

 this integral becomes 



m 2 a v h 



and the part which is independent of the initial state 



From this it is evident, first, that after an indefinitely long 

 time the exponential vanishes, and v in the limiting case is ex- 

 pressed by the sum of which it has been above remarked that 

 it becomes 



- W1 ph+1 - WQ h > {W) 



as had resulted from the direct consideration of the stationary 

 state. 



If at any time <t a change of the velocity w^ affecting the 

 surface occurs to the amount of +y, then is 



(f>(X) = Wi from \=0 to \=#, 

 (f>(\) = a\ + 7 from \= 6 to \ = t ; 



consequently the above integral divides into two and has the 

 value 



m l a, v ' ma v ' 



The second member, which accordingly is added to v, repre- 



f 



