308 Mr Harrison, The pressure in a viscous liquid 



These may be written 



m2 = (1 + uJ2v)l{l + B), 

 'I + uJ2v\^ J 1 + F 



sn' 



(^-^1?)-^ 



3F (1 + 2i//wo) ' 



If the values of Uq and a be given, the last equation serves for 

 the determination of h. Writing h-y = l/k, the equation has the 

 same form in k-^ as in Jc. Hence, if k is a solution, l/k is also a 

 solution. Therefore, of real values of k, it is only necessary to 

 consider such that satisfy ^ ^ ^ 1 . 



Treat a as small, and assume that ( — = — %-^ — ) a is also small. 



V l + k^ J 



(1 + 7(^2)2 



We have a^ = ^-.^ .^ \, , — tk— . • 



3P (2 + 2v/uq + uJ2v) 



The least value of a for a given value of Uq/2i', if k is real, is given 

 hj k = 1. In this case, if Uq/2p = 1, a^ = ^, a = -58. This value of 

 a is not small enough for the approximation to hold good. Put 

 ^ = 1 and 2v/uq = 1 in the original equation, and we find a = "65, 

 approximately. For smaller values of a, k will be a complex 

 imaginary quantity. As uJ2v is either increased or decreased, a 

 real value for k can be obtained for smaller values of a. 



It will be found sufiicient for the purposes of the present paper 

 to restrict the consideration of the solution to the ranges of values 

 of a and Uq/2v for which k has a real value. We proceed to discuss 

 the pressure variation in the case for which k is real ; the variation 

 in the case for which k is complex can be inferred by considerations 

 of continuity. 



Let J) be the mean pressure at the point (r, 9) in the liquid, and 

 p its density. We obtain from, the two-dimensional polar equations 

 of motion 



u^ _ Idp V d'^u 

 ~^^ ~pdr^r^W' 

 _ \ dy 2vdu 



^'^^^ ^-~2^-2^aP + ^(^) 



substituting for ^^ from the differential equation satisfied by u. 



Also ^=J^+/(y) 



p "'^ 



