PART II : HYDRODYNAMICAL THEORY OF TEMPERATURE OSCILLATIONS. 633 



i.e. on ff(p—p f )—- If we now consider the case of gradually varying density, we see 



that the force on which the motion depends is a — \ J-^-.-f'dh where H is the depth 



^ J dx J u \dh dh\ 1 



of the lake. By imposing a condition that at any particular depth there is no horizontal 



motion, it is seen that an infinite number of modes of oscillation are possible, but in 



a liquid of gradually varying density irrotational motion is impossible.'" Considering 



the case of a liquid where the discontinuity is sharp though not quite abrupt, it follows 



from what has been said that the period of the principal oscillations will be slightly 



Fig. 4. 



longer than our theory gives. If vortex motion is caused by slipping at the boundary, 

 this will also consume energy and tend to lengthen the period. 



§ 30. It will be instructive to consider more fully the case of a lake of rectangular 

 cross-section whose temperature normal curve is parabolic. The longitudinal section of 

 such a lake is shown on fig. 4. If h(x) is the depth of the lower layer at any point 

 x and h' the depth of the upper layer, adopting Chrystal's notation (H.T.S., p. 621) 

 and considering only uninodal oscillations, we arrive at the equations 



t A(l-w*) . 



(. 2Aw . ,, v 

 £= sin n^t - r) 



* Burnside, Proc. Lond. Math. Soc, xx. p. 392 (1889). 



(15) 

 (16) 



