It was shown long ago by Airy that when waves advance over shallow water of depth originally uniform, the crests tend to gain upon the hollows, so that the anterior slopes become steeper and steeper. Ultimately, if the conditions are favourable, there formed what is be may called a bore . Ordinary breakers upon a shelving beach are of this character, but the name is usually reserved for tidal bores advancing up rivers or estuaries. Interesting descriptions of some of these are given in Sir G. Darwin’s ‘Tides’ (Murray, 1898). Although the real bore advances up the channel, we may for theoretical purposes “reduce it to rest ” by superposing an equal and opposite motion upon the whole water system. We have then merely to investigate the transition from a relatively rapid and shallow stream of depth l and velocity u to a deeper and slower stream of depth l' and velocity u' (fig. 1). The places where these velocities and depths are reckoned are supposed to be situated on the two sides of the bore and at such distances from it that the motions are there sensibly uniform. The problem being taken as in two dimensions, two relations may at once be formulated connecting the depths and velocities. By conservation of matter (“continuity”) we have lu = l'u' . (1) And since the mean pressures at the two sections are ½ gl , ½ gl' , the equation of momentum is lu ( u - u' ) = ½ g ( l '² - l ² ); (2) whence u ² = ½ g ( l + l' ). l' / l , u ² = ½ g ( l + l' ). l / l' . (3)