422 The late Lord Rayleigh on 



f being written for </f/dR ; and 



Dm 



_a^_,a_, (f+¥)= _ t>x , 



D » TT/Vl^-L ^^f' X2 \x ?' X " « 



Hence 



J*=.Ar+2«.{X + {*X > i^=«^_2«(U-6r)+{V,. (6) 



and on integration 



£ =i G > 2 (^ 2 + y 2 )-2«U^ + j(2af + ?»)RdR. . (7) 



As might have been expected, the last term in (7) is the 

 same function of R as when U = 0, but R itself is now a 

 function of U and t. 



In the case considered by Sir N. Shaw, f is constant and 

 may be removed from under the integral sign. Thus 



P = ia> X^+f) -2a,Vy + (»?+ KW + (*-U*)»}. (8) 



r 



If U = 0, R 2 identifies itself with aP+y 2 , and we get 



p/ P =i^+m^+y 2 ) (9) 



A constant as regards x and y, which might be a function of 

 t, may be added in (8) and (9). 



We see that if © + f=0, that is if the original terrestrial 

 rotation is annulled by the superposed rotation, p is constant, 

 the whole fluid mass being in fact at rest. It was for the 

 purpose of this verification that the terms in &> 2 were retained. 

 We may now omit them as representing a pressure inde- 

 pendent of the motion under consideration. In the strictly 

 two-dimensional problem there is a pressure increasing 

 outwards due to "centrifugal force." In the application 

 to the earth's atmosphere, this pressure is balanced by a 

 component of gravity connected with the earth's ellipticity. 

 Thus in Shaw's case we have 



= Const. + «+ J? 2 ) { (y- ^pi^) 2 + (*- °0 2 } , (10) 



