CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 443 



there may be curve-factors which do not possess an order at infinity in this 

 sense. 



It is important to bear in mind that the form of the curved side of the polygon 

 which corresponds to a particular curve-factor depends not merely on the analytical 

 form of the curve-factor, but also on the Schwarzian or other factors which appear in 

 the formula of transformation. Thus the same curve-factor leads to different curves 

 in different transformations. But these curves will have some features in common, 

 provided the linear range with which the particular curve-factor is associated is not 

 also in whole or in part included in the range of some other curve-factor ; for example 

 if ( 0> in any such simple connexion represents a cm-ve everywhere convex to the 

 relevant region, then in other such connexions both ^ and positive powers of *$ will 

 always represent curves convex to the relevant region. This appears from considera- 

 tion of the vector angle of $*, which, when added to the vector angles of the other 

 factors, the latter being constants for the linear range of" *$, gives the angle of 

 direction of the tangent to the curve. 



6. Convention as to Fractional Power*. In the course of the work expressions 

 involving square roots and other fractional powers of quadratic expressions in tr will 

 occur frequently. The convention must therefore be made at the outset that when 

 X is fractional an expression such as (vf ) x shall be interpreted as having a positive 

 real value for real values of w greater than a, and for other values of w corresponding 

 to points on the relevant (positive) side of the axis of w real such value as corresponds 

 to continuous passage from a point where w is real and greater than a without 

 departure from the relevant half-plane, the branch point if = a being avoided when 

 necessary by a detour in the relevant region. With this convention an expression 

 such as (w ) A will be free from ambiguity, and in particular the passage round the 

 point a from a real value greater than a to a real value less than a will lead to the 

 form (awY exp(iX-Tr) for real values of w less than a. 



7. The Curve-factor of Semi-circular Type. It is natural to expect that the 

 study of the two curve-factors employed by Mr. PAGE may lead to suggestions for 

 the construction of other curve-factors. That which occurs in the transformation (l) 

 above may be called the curve-factor of semi-circular type ; it is 



^ = w + ^'-c 2 ) 1 '', (4) 



and has linear range from +.c to -c, and angular range *. 



It is to be noted that the vector angle of ^,, when c > w > -c, is tan" 1 [(c a -w a ) v '/w] 

 and it is only because the denominator of the fraction in the square bracket vanishes 

 for a point in the linear range that the angular range is TT instead of being zero. 



The reason why ( @ l satisfies the requirement of having no zero is easy to see. 

 Let Sf t be the surd conjugate to tf lt that is w-(iv*-c*}\ Every zero of ^ must 

 (since 9 l has no infinities) be a zero of the product V&, but tf^ = c 2 and has no 

 zeros, therefore < ^ > 1 has no zeros. 



