216] GENERAL CONDITIONS. 371 



To find the condition which must be satisfied at the free 

 surface (p = const.), let the origin be taken at the undisturbed 

 level, and let Oy be drawn vertically upwards. The motion being 

 assumed to be infinitely small, we find, putting fl = gy in the 

 formula (4) of Art. 21, and neglecting the square of the velocity (q), 



(3)- 



Hence if rj denote the elevation of the surface at time t above the 

 point (x, 0), we shall have, since the pressure there is uniform, 



provided the function F(t}, and the additive constant, be supposed 

 merged in the value of d<f>/dt. Subject to an error of the order 

 already neglected, this may be written 



*-JL?L < 5 )- 



Since the normal to the free surface makes an infinitely small 

 angle (drj/dx) with the vertical, the condition that the normal 

 component of the fluid velocity at the free surface must be equal 

 to the normal velocity of the surface itself gives, with sufficient 

 approximation, 



d*_ wi (6) 



This is in fact what the general surface condition (Art. 10 (3)) 

 becomes, if we put F(x, y, z, t) = y rj, and neglect small quanti- 

 ties of the second order. 



Eliminating ?? between (5) and (6), we obtain the condition 



to be satisfied when y = 0. 



In the case of simple-harmonic motion, the time-factor being 

 condition becomes 



242 



