604 Messrs. Cowley and Levy : Method of Analysis suitable 

 Now, round r — bja 



f,4 (V ^)=f 2 "^[A ( v^)] ; 



J e OK Jo a L 0^ -Jr=b/a 



Inserting these four conditions in the expression (18),. 

 for yfr we derive finally 



*> = 2(a'-ty«') P-S 1 °g^ + 1 ]- 

 Hence the angular velocity at any point is 



n _ 1 'djrp _ a 2 a> I _1S\ 

 11 ~r or " a 2 -h 2 \ R 2 /' 



where R is the actual distance from the centre to the point 

 (of. Lamb's 'Hydrodynamics,' § 333). 



Justification of expanding in powers of C near = 0. 



The foregoing analysis is based fundamentally on the 

 assumption that a convergent expansion for i/r in ascending 

 integral powers of valid over at least a finite range from 

 C = is possible. 



It has been customary in dealing with problems of fluid 

 motion to impose tentatively on the equations the restriction 

 that the inertia terms are negligible for slow motions. In 

 effect this is equivalent to a neglect of the second group of 

 terms in (8), the assumption being that if a solution of 



\7 4 >|r = can be found, the inertia terms ~^ ~- V 2 ^ etc. 



x By 



will at most be of the same order as V 4 ^ ; and since C is 



small in comparison with unity, that group of terms may be 



neglected. That this is justifiable a posteriori is clear 



when we remember that, on the analogy of the flat plate, 



the equation 



V^o = o 



will represent the deflexion of an unloaded flat plate where 

 the boundaries are compelled to satisfy certain conditions as 

 regard slope etc. In order that the analogy may be valid, 

 these slopes, although reckoned as of the order of standard 

 slopes, are small. It follows that all the derivatives of ^ 

 which occur in the equation are approximately of the same 

 order, and therefore those involving the factor in the 

 equation may be neglected in comparison with V 4 ^- On 



