252 
MR. S. S. HOUGH ON THE A.PPLICATION OP HARMONIC 
surrounding the points in question. It seems most probable that the inclusion of the 
terms involving the squares of velocities would have the effect of diverting the stream¬ 
lines, so as to cause them to follow the coast-lines even in the immediate neiohbour- 
hood of the equator. 
An attempt to trace out the lines /i/p, = const for the North Atlantic Ocean from 
data obtainable from the Admiralty charts quickly showed that the theory here put 
forward is inadequate in itself to explain even the more salient features of the 
circulation in the region in question. Observation however indicates the direction 
in which we must look for the defects of this theory. The excessively low temjDera- 
ture of the water beneath the surface'" in equatorial regions can only be explained 
by supposing that this water has travelled thither from higher latitudes, whereas we 
know that the currents at the surface, for the most part, set from the equator towards 
the poles. We conclude that the under-currents of the actual ocean differ materially 
from the surface-currents, and in this respect the actual circulation differs from the 
types of circulation with which we have hitherto been concerned and which are the 
only possible types of circulation in our ideal ocean in which the density is uniform. 
It has been urged by some authorities that the variations in the density of the water 
arising from differences of temperature, salinity, &c., are the sole causes which 
maintain ocean currents, but in that we have seen that currents could exist even 
without such variations, it seems to me to be highly improbable that such is the 
case, though there can be no doubt, in the-light of our present analysis, that these 
variations are largely effective in determining the course which the currents pursue. 
If we suppose that the ocean consists of a number of horizontal layers of different 
densities, but that the density throughout each stratum is uniform, then for each of 
the strata a function xfj will exist defined by 
xjj =iY' -f y^) — y -f- const, 
and the horizontal velocities for any stratum will be connected with the corresponding 
function by the equations (34). The equation of continuity for any stratum may be 
formed as in § 3, provided we replace h by the depth of the stratum in question 
instead of the depth of the wiiole ocean. The stream-lines for any stratum will 
therefore still be given by the equation (38), with this modification in the meaning 
of the symbol h. It follows that the equator will still be one of the free stream¬ 
lines, but the motion elsewhere may be totally different from what it would be if the 
density were the same throughout. It seems probable then that the result we have 
obtained with reference to the tendency of the currents to set along the equator will 
still hold good even when the density is variable, and this conclusion is borne out by 
* The principal facts at present known in relation to the distribution of toinperatnrc in the ocean 
will be found in the ‘ Report of the Qhallewjer Scientific Results,’ ‘ Chemistry and Phjsics,’ yol. 1. 
