CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 477 



Adverting for example to 14, it is seen now that the provisional exclusion of a 

 negative value of ny was an unnecessary piece of caution. For ny negative the flow 

 would take place in a region like that on the 

 left of the arrows in fig. 16. It is true that 

 in this flow the velocity at infinity would be 

 infinite, but then there is no irrotational con- 

 tinuous flow possible in the region which does 

 not have infinite velocity at infinity. The / ~ c 

 flow is, in fact, no more impossible than is the 

 region in which it is supposed to take place. 



43. Curve-factors not having a definite order 

 at infinity. -An example of a curve-factor of a Fig. 10. 



kind not likely to be directly useful in physical applications is 



/M+grM^-c 8 )'''}, (H,y) 



where f and g are rational algebraical functions, or any functions which have no 

 infinities other than for w infinite. The vector angle, on the linear range < > ir > c, 

 i s X = 9 ( w ) (c a to 2 ) 1 '-, and the angular range is zero. 



^.., 7 has usually no definite order at infinity, and so the proposition of ijll dues not 

 apply to it. 



The possibility, however, of *$ m having a definite order at infinity may be 

 illustrated by an example which is in one respect a little more general than ' 37 , 



namely, 



w-c-(w-aY(w-by-*}. (110) 



For iv great this tends to the limit exp{a + (l a) b c}, so that the order at 

 infinity is zero. 



Of somewhat similar character are the curve-factors 



ay(w-by- a -(w-aY(w-by- 1 '}, .... (ill) 

 l-a)b + c\ w-(w-c](w-aY (w-b} 1 -]. . (112) 



CURVE- FACTORS KEGARDED AS THE LIMITS OF PRODUCTS OF SCHWARZIAN FACTORS. 



44. The Schwarz-Christoffel transformation being so widely known and of such 

 proved utility, the most natural way of trying to obtain a transformation for the 

 conformal representation of a region whose boundary is partly curvilinear would 

 seem to be to treat the curve as the limit of a rectilineal polygon and to seek the 

 corresponding limit of the product of the Schwarzian factors appropriate to the 

 corners that have to be smoothed out into a continuous curve. 



