598 Messrs. Cowley and Levy : Method of Analysis suitable 



be more conveniently analysed. Reverting to the experi- 

 mentally known facts that the value of: UL/v determines for 

 any particular problem the nature of the flow, we see that 

 this finds its counterpart in the presence of that quantity as 

 a variable parameter C in the differential equation. It is 

 clear, in fact, that mathematically as well as physically the 

 whole problem may be made to centre round this parameter; 

 and since we are more directly concerned in the modification 

 in the state of the flow over the whole field as it were as C 

 varies, rather than with a comparison with the state of flow 

 at one point in the a\y plane with that of another, the most 

 natural form of solution that is suggested would be a solution 

 as a series in powers of C. Whether or not such an expansion 

 is possible as a convergent series is a matter which will be 

 entered into shortly, but for the moment it is proposed 

 to assume that the expression for yjr the stream-function 

 may be written in the form 



yjr = ^ + ^ 1 C + 'f 2 C 2 + (14) 



[It may here be remarked that if the problem of the motion 

 of a body had been solved for a given value of 0, the solution 

 for the same body moving backwards may b« derived by 

 writing -C for C in this expression.] Inserting this in 

 equations (8), (10), (12), etc., and equating the coefficients 

 of each power of C to zero, since the equations hold for all 

 values of that variable, the following system is obtained : — 



V 4 ^o = 0, 

 ofo _ A 

 0% 



nd 



oy 





round the moving body 

 ?3 



► Coefficient of c°. 



V 4 t: = 



Aody° 







OJr< 



oy ox 



0, anc 



V 2 fo" 



■ 3*i 



ox ' "' ~~ oy 



round the boundaries 



O'V oy 

 = 



V 2 tc 



I 



I . .. 



r Coefficient of 



. (8«) 



. (10 a) 



. (12a) 



(86) 



(10 b) 



| ds 



«■ body 



, on 



-i 



body on On ' 



(12 b) 



etc., etc. 



In the above the expanded expression for yjr has been inserted 

 .not merely in the differential equation, but also in the integral 

 <-md boundary conditions, and in all cases the coefficients of' 



