476 DR. J. G. LEATHEM ON SOME APPLICATIONS OF 



Integration leads to the approximate relations 



w = Bz"''^ + C, dw/dz = DTrft~ z ~ , 



where B and C are real constants. 



Now if w be expressible, for great values of z, as a series of terms of the type cz\ 

 since iv has- to be real for z real, c must be real ; and since w has to be real for z of 

 the form r exp (ift), it is necessary that X/3 = mr, where n is a positive or negative 

 integer. Thus the admissible terms in the expansion are of the type c n z mlft . When 

 ft = TT the expansion may also include a term in log z, but it must be remembered 

 that z, when great, is of a higher order of greatness than log z. 



The real part of w being represented by <p, it is known that 



taken over any area in the z plane is equal to <f> (30/3/) ds taken round the boundary, 



J 



?v representing an element of outward normal. If the boundary be made up of the 

 locus of ir real and an arc of a circle with centre at the origin and radius r, the 

 subject of the line-integration is zero on the former part of the boundary. On 

 the latter part Or rr and cZ.s = rdQ ; thus 



I liC<l>\ 2 , /C0\ 1 77 I" 



U ^- + Uf dx dy = 

 JJ l\3ay ^ oy] Jo 



where a -> /3 as r is increased indefinitely. If the area-integral is to cover the whole 

 of the relevant region the limit of the right-hand side must be taken for r -> <. 



A term c n z nnjf> in w would give in the limit of the line-integral a term ^mrc n 2 r 2nir ^, 

 and this tends to zero if n is negative and to infinity if n is positive. If the area- 

 integral vanishes w must be constant throughout the region. Hence, for an assigned 

 value of ft, there can be no w of any significance unless the most important term 

 in iv for z great has a positive index (or, as a possible alternative in the case of ft = TT, 

 is a multiple of log z). For this condition the least admissible value of n is unity. 



Thus generally the transformations made up of curve-factors and Schwarzian 

 factors are such that the most important term in w, for z great, is of the least possible 

 order of magnitude that is consistent with w being other than a mere constant. It 

 is true that, in the case of ft < TT, div/dz is infinite for z infinite, but the conditions 

 of the problem do not then admit of any w free from this objection. 



In the hydrodynamical applications it may be said that the transformation gives, 

 for any specified region in the z plane, an irrotational continuous motion which is as 

 free from singularity at infinity as the nature of the geometrical configuration allows. 



