208 
:\]R. S. S. HOUGTI ON THE APPLICATION OF HARMONIC 
the difficulties arising frnm this cause ; the fluid which constitutes the ocean may be 
regarded as a thin layer distributed over an approximately spherical surface, a 
circumstance which enables us to reduce the number of our independent variables 
and to treat the problem as a two-dimensional one. 
Before proceeding to the transfoi-mation of our equations, let us examine them in 
the form in which they are given above. If we suppose the system is executing 
a simple harmonic vibration in period 27r/'X, we may put v, iv, ifj each proportional 
to and therefore replace du/dt, &c., by iku, &c. Thus the equations (2) give 
iku — 2 ojv 
iXv + 2(jju 
i\iv 
dip' 
d.v 
dip- 
vy 
dip- 
dz' 
Now in an important class of oscillations, viz., the tides of long period, the value 
of X will be small compared with that of oj ; while for another class of motions, viz., 
the steady ocean-currents, we must suppose X absolutely zero. In these cases, if we 
retain only the most important terms, the equations of motion take the approximate 
form 
— 2o)V = 
dip 
d.v ’ 
Hence, applying the operators d/dz to the first two, and making use of the third, 
we find 
dujdz = 0, dvjdz = 0. 
Likewise also from the equation of continuity, 
dw dii dv _ 1 / d~ip d-ip \ _ 
dz dx dy 2co \dx dy dy dxj 
From this we see that in the case of tides of very long period the velocity of the 
fluid particles is approximately the same at all points in the same line parallel to the 
polar axis, while in the case of the ocean-currents this is rigorously the case. 
Now in order to effect the transformation of the equations of motion, it has been 
assumed by Laplace and his followers that, on the analogy of “ long waves ” when 
there is no rotation, all fluid particles which are at one instant in a vertical line 
will remain in such a line. This assumption appears to require some modification in 
the case of our rotating system. We shall see hereafter however that the assump¬ 
tion in question will not lead to appreciable error, provided that the depth of the 
