630 MR E. M. WEDDERBURN ON THE TEMPERATURE SEICHE. 



being uniformly p' and below p. Let the origin be taken in the surface of separa- 

 tion between the two layers of water, and consider any cross-section of the lake at a 

 distance x from the origin measured along the longitudinal axis, which, as in the case of 

 ordinary seiches, should be as nearly as possible in the average direction of the channel 

 of greatest depth. Let the area of the section of the upper liquid be A'(x) and of the 

 lower liquid A(x), and the breadth of both at the surface of separation b(x). The 

 volume of slices (S r and S) at that point will be to the first order of small quantities 

 A\x)dx and A.(x)dx* 



§23. Suppose that after a time the slices S' and S have moved into new positions 

 so that the distances of the posterior faces from are respectively x — %' and x + £. 



Then the thickness of S' in its new position will be dx(l — -£ ), and of S, dxl 1 + — ). 

 The volume of the part of S' above the surface of separation will be A' (a; — ^)dxi 1 — — ), 



and of S below the surface of separation, A(x + £)dx(l + — )• It is now assumed 



that the alteration in the level of both slices is the same throughout the slice, 

 and is the same in both slices, viz : — £. This is not quite accurate, as the slices move 

 in opposite directions, but the assumption only involves quantities of the order 



m=Sl, which is very small. It is further assumed that there is no transverse motion, 



and that all the water particles in the same transverse vertical place (in each liquid 

 separately) have the same horizontal velocity. This involves slipping at the surface 



of separation, t and also that ( -A is a negligible quantity .| 



§ 24. We may then take the change in volume of the two slices due to a rise £ of 



the water of the lower liquid above the surface of separation to be —b{x)Xdx(l—^A 



for the upper liquid and b(x)^dx( 1 + -j) for the lower liquid. 

 The equations of continuity for the two liquids then are 



A'(x)dx={A'(x-^)-l(x)^}dx(l-~) .... (4') 



A(x)dx={A(x + £) + b(x)Z}dx(l + d ^ . . . (4) 



that is, 



&(x)=-A'(x)l(l- d ^ + A'(x-?) .... (5') 



= A(x)l(l+^-A(x + £) . . . (5) 



* Throughout I have followed Chrystal's notation and method closely, 

 t Lamb, p. 354. + lb., p. 244. 



