ON TEMPERATURE OBSERVATIONS IN LOCH EARN. 645 
HyprRopYNAMICAL THEORY OF TEMPERATURE OSCILLATIONS. 
In the discussion of the ‘“ Hydrodynamical Theory of Temperature Oscillations,” 
which appeared in the Transactions, vol. xlvii., Part IV. p. 628, the following assump- 
tions were made: (1) That the amplitude of the oscillation was small, (2) that there was 
no transverse motion of water particles, and (3) that there was an abrupt discontinuity in 
temperature at a certain depth, above and below which there was uniform temperature. 
In what follows, this last assumption is removed and the case of oscillations in a liquid 
of gradually varying density is considered. In such a case irrotational motion is not 
possible, but as the amplitudes with which we are dealing are small, and the 
velocity of water particles is slow, we shall only be neglecting very small quantities if 
we assume that the motion is irrotational. 
We have seen (2bid.., p. 633) that by imposing a condition that at any given depth there 
is no horizontal motion, an infinite number of modes of oscillation is possible. Let us then 
consider a transverse vertical section of the lake at a horizontal distance «, from a plane 
perpendicular to the axis of the lake, which we shall call the plane of origin. As before, 
the axis of the lake is taken as following the average line of greatest depth. Assume 
now that at a certain level there is no horizontal motion, which is equivalent to assum- 
ing that above and below this level the motion of the water particles is in opposite 
directions. Let the area of the whole cross-section be {A’(~)+A(a)', A’(a) being the 
area of the section above the level at which there is no horizontal motion, which we 
shall call the boundary, and A(a) the area of the section below the boundary. Further 
consider A’(x) split up into thin lamine A’,(«), A’j(x), A’,(~) ... and A(x) into 
A,(x), Ap(a), Aj(a). . . . The volume of slices S’ and S above and below the boundary, 
with thickness dx at the point under consideration, will then be respectively 2A’,(x)dx 
and 2A, (x)da. 
Suppose that, after a time, the slices 8S’ and S have moved into new positions, so that 
the distances of the various laminz from the plane of origin are (w—,), (~w—&,), 
(a—&,) . . . for those above the boundary, and (w+ &), (w+ &), (w+&) . . . for those 
below. Then the thickness of the various laminz in their new positions, above and 
7 
below the boundary respectively, must be dx ( — =) and da (1 == - . The volume 
of the slice will then be 
BA’ (0 £y)de(1 - ) 
and of § 
BAy(e + &)de(1 +2), 
It follows from the assumption that there is no transverse motion, that the change 
of level of any lamina is the same throughout its whole breadth. Let ¢,, Co, Vg... 
GC» Ce Gs - - - be the difference between the change of level of successive laminz, as 
S’ and S move to the new position, so that =’; = 2%, =(, where ¢ is the change of level 
TRANS. ROY, SOC, EDIN., VOL. XLVIII. PART III. (NO. 26). 94 
