[62] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION, &c. 



pressure on a plane in the direction of dS. Then by the formulae (9) of my former paper 

 applied to the equations (142), (143) we get 



*P-- g Jr^ + »'-^ + «'-££4 f .... (144) 



[ dor doe dy dot dx) 



with similar expressions for A Q and A R, and A P, A Q, A R are the components of the 

 pressure which must be applied at the surface, in order to preserve the original motion 

 unaltered by friction. 



53. Let us apply this method to the case of oscillatory waves, considered in Art. 51. 

 In this case the bounding surface is nearly horizontal, and its vertical ordinates are very small, 

 and since the squares of small quantities are neglected, we may suppose the surface to 

 coincide with the plane of xx in calculating the system of pressures which must be supplied, 

 in order to keep up the motion. Moreover, since the motion is symmetrical with respect to the 

 plane of xy, there will be no tangential pressure in the direction of *, so that the only 

 pressures we have to calculate are AP 2 and AT 3 . We get from (140), (142), and (143), 

 putting y = after differentiation, 



AP 2 = - 2/u.m 2 c sin (mx - nt), A T% = Z/itrfc cos (mx - nt). . . (145) 



If U|, «i be the velocities at the surface, we get from (140), putting y = after differen- 

 tiation, 



«i = mc cos(mn - nt), u, = - mc sin (mx - nt). . . . (146) 



It appears from (145) and (146) that the oblique pressure which must be supplied at the 

 surface in order to keep up the motion is constant in magnitude, and always acts in the 

 direction in which the particles are moving. 



The work of this pressure during the time dt corresponding to the element of surface 

 dx dx, is equal to dx dx (A T 3 . u x dt + AP, .v x dt.) Hence the work exerted over a given 

 portion of the surface is equal to 



2/uim 3 (?dt ffdx dx. 



In the absence of pressures A P 2 , A T 3 at the surface, this work must be supplied at the 

 expence of vis viva. Hence 4/fim 3 c ! dt ffdx dx is the vis viva lost by friction, which agrees 

 with the expression obtained in Art. 51, as will be seen on performing in the latter the 

 integration with respect to y, the limits being y = to y = oo . 



