CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 475 



^as is really the product of two simpler curve-factors, one of the now familiar type 

 of ^ 13 , the other 



a double curve-factor whose linear ranges are a > w > and > w > co , having 

 total angular range zero, and modulus unity for w < 0. 



The formula of integration by parts suggests that other forms of ^ M could be 

 factorised in a corresponding manner. 



41. A configuration of the type of fig. 15 is given by the transformation 



, A (a/-+ w ''')*- (6'/ W')" {(-&)/*+ ( jt ,-7,yH 

 ( w -aY(w-V)" 



where a > b > 0, n is arbitrary, and PTT, qir are the angles indicated in the figure. 

 The boundary corresponding to w negative is a free stream-line. Greater generality 



can be obtained by introducing factors of the typo {(>.<> c)''-+ (' c)' /2 i , wliere 

 6> c> 0. 



GENERAL REMARKS ON CURVE-FACTORS. 



42. Note on the Relation between Angular Range and Order at Infinity. It has 

 been already shown that the angular range of a curve-factor is x times its order at 

 infinity ; the same is obviously true for a Schwarzian factor. Hence all the trans- 

 formations which have been considered are characterised by the property that the 

 total angular range is TT times the order at infinity. Thus if the angular range be 

 /3 TT, where TT S ft > 0, the first approximation to the form of the transformation 

 for values of w having very great modulus is 



dz = AttA'- 1 dw, 



where A is a real constant, and the limit form of the boundary consists of lines 

 constituting the arms of an angle /3. 



