Leathkm — On Two- Dimensional. Fluid Molina. 17 



that ('„ belongs to the type of ('-. but is closely akin to ff n and has zero 

 order at infinity. 



Still another type of sub-factor is the foundation of the formula of G, §59. 

 It is believed, however, that the one now to be discussed is simpler and more 

 useful for developing general theory. 



7. Suggestion is found in the ohvious device of taking the fixed curved 

 boundary of the obstacle .-is the limit of a rectilineal polygon, for which latter 



ty] f obstacle the solution of the hydrodynamical problem is, in one form 



or another, well known. Willi very different analysis Cisotti paper B) has 

 dealt with a polygonal obstacle, and Yill.il paper C) with a curved obstacle 

 as the limit of a polygon. In paper I», §.'18, the present writer has outlined 

 the formulation of this process in terms of curve-factors, for the particular 

 case of symmetry, but without such demonstration of the generality of the 

 formula as is necessary if further theory is to he built upon it. 



Beginning with the asymmetrical case, it is convenient to replace / ,('^ 

 by (£-cJ C'{Z), s " that the geometrical relation is 



J-. = (Z-c)C{Z)iK t7) 



Then, since Z - c has angular range tt, what is required of £(Z), besides 

 generality, is that it be a curve-factor of angular range zero, such that for all 

 real values of X, greater than a or less than - a the modulus of £(JL) is unity. 

 Lest the ultimate parallelism of the free stream-lines should be regarded as 

 an unjustified assumption, it may be recalled in passing that the angular 

 range of a curve-factor is always tt times the order at infinity (D, § 5), so that 

 the constancy of | (?(£) | for Z ->- w is not compatible with any angular 

 range for Git,) other than zero. 



Now if in £«(£,), as defined in equation (5), « = (3 = a, and r is replaced 

 by a real parameter k intermediate in value between - a and a, there 

 results a curve-factor 



' ,..U-, Z) - - / {kZ - a?) (^ - ic 1 ) - * + (P - «')*, < s ' 



whose angular range is n and whose modulus, for Z real and Z~ > "", is 



Hence £ti(k, £)/(£- k) is a curve-factor which, if multiplied by a suitable 

 complex constant, satisfies the particular requirements of i ^ ; and any 

 power of this expression is equally suitable. Further, for Z real and Z 1 > «~. 

 the vector angle of the expression is constant, save for abrupt change in 

 passing through the value Z = K. 



