liquid contained between two rotating coaxal circular cylinders 457 



Thus equations (23) are replaced by 



a'^ _ 52 _ 2^2 _ 2 (1 + X^)A 



(^2 + ct'2) (p2 _ 52) = (1 _ A2)2, I (3). 



2p (a'2 + s^) = -k. J 



Expressing a' and s in terms of p, we have 



s2 = 2 V{A2 + (1 + A^) ^2 + p^} - (1 + A2 + :p2) ...(4), 

 a'2 = 2 V{A2 + (1 + A^) p^ + i)^ + (1 + A'- + :p') •••(5)- 

 Thus ^ = - 4^ V'{A"' + (1 + A2) i?^ + ^4}^ 



Now j9 may be taken either positive or negative, and A is either 

 positive or negative according to the nature of the motion of the 

 cylinders. Putting 



. a%^oj ,, . 



^ = p^i?' e- > «) 



and assuming p and to to be treated as positive, we have 



P^l^h- ^^ ^ V{A^ + (1 + A2) p^ + p'} (7). 



The discussion of equation (2) proceeds exactly as in Orr's 

 paper, using the values for s and a' obtained above, and it is easily 

 shown that there is no solution for which sn is less than tt except 

 one for which sn is zero. 



Putting s^ = p^ + ein (4), we see that 



(1 - A2)2 + 2e (2^2 + 1 + X2) = 0. 

 Hence e is negative, and p is always numerically greater than s. 

 Thus from (5) a' > V^p > \/3s, and, therefore, sinh a'n and cosh a'n 

 each exceed 100 and are approximately equal. Thus (2) becomes 



tan sn - ^^^ ^ ^,^ _ ^2 



_ Vi^P' + 2(1+ A2) jt)2 - (1 - A2)2} 

 ~ 3pMnr+A2 



We have to solve the equations (4) and (8) for p and s, using 

 integral values of A. The result of the substitution in (7) gives the 

 corresponding critical values of ^u. Now for a given A, tan sn given 

 by (8) has no real stationary value, and thus tan sn must be less 

 than Vi Hence the smallest value of sn, except the zero value, 

 satisfies the inequality 77 <sn <77r/6. The corresponding value of 

 p is also the smallest of the series of roots in p, and therefore gives 

 the left-hand side of (7) its minimum value for given values of b/a 

 and A. 



