648 MR E. M. WEDDERBURN 
Making this assumption, we have 
o(e)m= nb le) + Bele) 
A@)* A(@) 
If there is an abrupt temperature discontinuity with uniformity of density of p’ above, 
and p below, this expression reduces to aa which is the form obtained in our 
Ke) * VC) 
preliminary investigation. 
There remains the consideration of the depth at which the boundary should be taken, — 
Theoretically, an infinite number of modes of oscillation is possible, but in practice 
we find that only one period occurs, or at least that the oscillations which occur in any 
loch have very nearly the same period. There is, therefore, an element of doubt as to 
the depth at which the boundary should be taken. So long as the motion of the water 
particles is irrotational there must be slip at the boundary, and the physical difficulty to 
such a slip can be overcome by assuming the formation of a thin vortex sheet at the 
boundary, and, as the velocities are all small, such a sheet might be so thin as to be 
negligible for the purposes of this discussion. If we consider a body of water of 
gradually varying temperature, in which the isotherms have, from any cause, been — 
inclined from the horizontal, but are now at liberty to return to the horizontal position, 
the force which is acting at any point is proportional to the rate of change of density at 
that point. Accordingly, if slip of one layer of water over another is to take place any-— 
where, it is most likely to occur where the density gradient is greatest. 
Very often, in nature, there is a considerable distance through which the density 
gradient is at a maximum, and in such a case probably the exact depth at which slip 
takes place may be determined by the viscosity of the water, and as the viscosity of 
water decreases more rapidly as the temperature rises, it is likely that if there is 
any distance through which the density gradient is at a maximum, the slip will occur 
where the maximum gradient commences. 
The foregoing theory does not attempt to be a rigorous mathematical treatment of 
the problem of standing waves in a heavy liquid of gradually varying density, and its 
usefulness will depend on the degree of accuracy with which it fits in with observed 
facts, but before applying the theory to the calculation of the periods of Loch Harn it 
will be well to summarise the assumptions made. Perhaps the greatest error is in the 
supposition that the motion is irrotational and that the effect of vortices, which are 
certain to be formed at the boundary, is negligible. We should expect the theory to. 
give too short a period for the oscillation on this account. It is also assumed that the 
amplitude of the seiche is small compared with the depth of the lake, but this assump- 
tion is also wide of the mark, for the amplitudes which we find are often a large 5 ; ‘< 
