414 Proceedings of Poyal Society of Edinburgh. [sess. 
of the second order, negligible in comparison with g£ v which is an 
infinitely small quantity of the first order. 
§ 121. For a sinusoidal wave-disturbance of wave-length 27 rjm, 
travelling towards with velocity v, we have as in (66) above, 
F(a? , ^ , t) = - ke~ m{z ~ 1] sin m(x - vt) . . (152). 
For surface-equation (151) becomes 
0 = kmv cos m(x - vt) - .... (153). 
This gives as the equation of the free surface 
Ci = h cos m{x — vt) ( 1 5 4), 
where h = - "" (155). 
9 
Now by (149) and (152) with z = 1, we find 
= — cos m(x — vt) (156). 
Comparison of this with (154) gives 
v 2 = glm = \gl'27r (157). 
§ 122. Let us now find A (activity), the rate of doing work by 
the pressure of the water on one side upon the water on the other 
side of a vertical plane ( x ). We have 
r oo v r oo 
A = J dzp£ = j dz£ 
d F 
dt 
+ g(z - 1 + C) 
(158). 
Eliminating from this £ and F by (149) and (152), we find 
km cos 
m(x - v t)j^ d Z€ m[z 11 ~ kmv* m(z 1} cos m(x -- vt) + g(z - 1 + C) 
(159). 
Hence, performing the operations dz , we find 
A = - km cos mix - vt) 
kv , / 1 C\ 
_ cos mix -vt) + g{ — + — ) 
2 \m 2 mj 
(160). 
§ 123. Remarking now that 27 r/mv is the periodic time of the 
wave, and denoting by W the total work per period, done by the 
water on the negative side of the plane ( x ) upon the water on 
the positive side, we have 
W = [ dt . A = — • \ • k^mv — \^h 2 . (161). 
/ mv 2 2 
