614 Lord Rayleigh on the 



In (6) the motion of the fluid, represented by IT simply, is 

 given independently of £*, and the equation is the same a* 

 would apply if £ denoted the temperature, or salinity, of the 

 fluid moving with velocity U. Any conclusions that we may 

 draw have thus a widened interest. 



In Kelvin's solution of (6) the disturbance is supposed to 

 be periodic in x x proportional to e iJiX , and U is taken equal 

 to fty. He assumes for trial 



£=T/{ 7 ^ +(w ~^ )y l", . . . . . (7) 

 where T is a function of t. On substitution in (6) he finds 



whence T=0 ,-^*+'*--W*H**>f} j . . . (8) 



and comes ultimately to zero. Equations (7) and (8) deter- 

 mine f and so suffice for the heat and salinity problems in an 

 infinitely extended fluid. As an example, if we suppose 

 n=0 and take the real part of (7), 



?=T cos A(.r- &.#), (9) 



reducing to f=Ccos&# simply when t — 0. At this stage 

 the lines of constant fare parallel to y. As time advances, 

 T diminishes with increasing rapidity, and the lines of con- 

 stant f tend to become parallel to x. If x be constant, f 

 varies more and more rapidly with y. This solution gives 

 a good idea of the course of events when a liquid of unequal 

 salinity is stirred. 



In the hydrodynamical problem we have further to deduce 

 the small velocities u f v corresponding to f. From (2) and 

 (3), if u and v are proportional to £***, 



Thus, corresponding to (9), 

 2T 



' = - *(i+ / 8y) Bin *(*-fl- y) - ' ' ' (11) 



No complementary terms satisfying d 2 v/dy 2 --7c*v=0 are 

 admissible, on account of the assumed periodicity with z* 

 It should be mentioned that in Kelvin's treatment the dis- 

 turbance is not limited to be two-dimensional. 



Another remarkable solution for an unlimited fluid of 



