PROM E OR ATE’ SU BAVQUEOUS "SHORE TERRACE 5 
time. The two particles have always the same phase and hence 
their movements are parallel at a given instant. The same is 
true of the third particle and all below it, the orbits decreasing 
ina descending geometrical progression (Fig. 7).7 
This fact is to be taken with one above stated, namely, that 
if orbits be decreased while the angular differential movement 
remains constant, the sharpness of the trochoid curve is reduced. 
It results from these properties that in a breaker where the curve 
of the surface would intersect itself, and is therefore impossible, 
the trochoids below the surface would show less of looping 
until a level is reached where normal wave motion is going on 
(compare Figs. 4, 5 and 6). 
Lines of like phase.— If the orbits of a vertical series of par- 
ticles be represented in diagram (see Fig. 7) and the correspond- 
ing points on the circles be connected with lines, then the line 
connecting the highest points and that connecting the lowest 
points of the several orbits are seen to be straight and vertical. 
The remaining lines are curved and inclined. In Fig. 8 these 
lines of like phase are shown in the positions where they occur 
in the wave. The particles ranged along any one of these lines 
would be ina vertical line if the water were at rest, just as all 
particles on one of the trochoid curves would le in a horizontal 
limes 
Consequences of the trochoidal form and of decreasing orbits 
below.—Ilf a horizontal plane be passed midway between the 
level of the crests and that of the troughs it will pass through 
the centers of the orbits described by the surface particles. All 
the water at the surface above this plane will then have a for- 
ward component, and all the water at the surface below this plane 
will have a backward component. An inspection of the dia- 
grams will show that the crests are steeper and shorter than 
TRANKINE, Joc. cif. p. 131. RUSSELL, “ Report on Waves made to the meeting 
of the British Association 1842-3,” reprinted in Zhe Wave of Translation, London, 
1885, gives formule adapted from Gerstner for the rate of orbitical diminution with 
depth. 
2 RANKINE, loc, cit., p. 129. 
