18 Lord Kelvin on the 



Supposing now the impulses to be at infinitely short 

 intervals of time, we translate the formula (170) into the 

 language of the integral calculus as follows : 



{.v, -, f) = [ \h]f(t-c,)-Y( X , z, q) . . (171). 



where f(t — g) denotes an arbitrary function of (t — §0, accord- 

 ing to which the surface-pressure, arbitrarily applied at time 

 (t—q), is as follows : 



U{t.-g)=-f(t-gyV(.v, 1,0) . . (172). 



Hence the pressure applied to the surface at time t, denoted 

 by Tl(x 9 1, t), is as follows : 



n(*,M) = -/«V(tf, i,0) . . (173). 



§ 129. The solution (170) or (171) gives the velocity- 

 potential throughout the liquid which follows determinately 

 from the dynamical data described in §§ 127, 128. From it, 

 by differentiations with reference to x and z, and integrations 

 with respect to t, we can find the displacement components 

 f, ? of any particle of the liquid whose co-ordinates were 

 .!', : when the fluid was given at rest. But we can find 

 them more directly, and with considerably less complication 

 of integral signs, by direct application of the same plan of 

 summing as that used in (170), (171). Thus if, instead of 



Y(x, z, q) in (171), we substitute j-Y(x } .;, g), and again 



7 m CLX 



-j-Y(x, z, q), we find f and f. And if we take 



W^V((B,*, q) and ("'<*?£ V(«, z, q) . (174) 



) t/ 



in place of V(#, z, q) in (171), we find the two components 

 f , J of the displacement of any particle of the fluid. Confining 

 our attention to vertical displacements, and using (179) below, 

 we thus find 



«*, *-, t)= I [dqfit-q) £-V{x, z, q) . (175). 



§ 130. To illustrate the meaning of the notation and 

 analytical expressions in (171), (173), (175), take the simplest 

 possible example, f(t — q) = l. This makes II the same for 

 all values of t ; and (173) becomes 



n=-V(*,i,0) .... (176); 



j: 



