738 Lord Kelvin on 



d = X -^co&m(x~-vt + 8) + Acos^(a:~-vt + y) (73), 



g — mv 2 v ?r v v " 



where B, m, /3 are given constants having different values in 

 the different terms of the sums ; and v is a given constant 

 velocity. The last term of (73) expresses, with two arbitrary 

 constants (A, y), a train of free waves which we may super- 

 impose on any solution of our problem. 



§ 41. It is very interesting and instructive in respect to 

 the dynamics of water-waves, to apply (72) to a particular 

 case of Fourier's expansion of periodic arbitrary functions 

 such as a distribution of alternate constant pressures, and 

 zeros, on equal successive spaces, travelling with velocity v. 

 But this must be left undone for the present, to let us get 

 on with ship-waves ; and for this purpose we may take as a 

 case of (72), (73), 



U=gc(i ^ e cos 6 + e 2 cos 20 + &c.)=gc—^—^—^ 2 (74), 



2j + j^t cos ^+ j3Tjcos20 + &c. I . . (75); 

 where 



0=^{b-vt+/3) (76); 



„ gX T a <ja 



t= W ; J =x = w < 77): 



and e may be any numeric < 1. Remark that when 

 u = 0, J = co, and we have by (75) and (74), &=U/g, which 

 explains our unit of pressure. 



§ 42. To understand the dynamical conditions thus pre- 

 scribed, and the resulting motion : — remark first that (74), 

 with (76), represents a space-periodic distribution of pressure 

 on the surface, travelling with velocity v ; and (75) represents 

 the displacement of the water-surface in the resulting motion, 

 when space -periodic of the same space-period as the surface- 

 pressure. Any motion whatever ; consequent on any initial 

 disturbance and no subsequent application of surface-pressure; 

 may be superimposed on the solution represented by (75), to 

 constitute the complete solution of the problem of finding the 

 motion in which the surface-pressure is that given in (74). 



§ 43. To understand thoroughly the constitution of the 

 forcive- datum (74) for II, it is helpful to know that, n de^ 

 noting any positive or negative integer, we have 



Mi + . - + e* cos 20 + &e.) = JT frl + ( ^_ wa)8 (78), 



