440 DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



At the outset something of a suggestion might seem to be found in consideration 

 of the fact that, if the intrinsic equation of a curve in the z plane is x = F (s), the 

 vectorial element of arc dz is the same as ds exp (?x) or ds exp {t'F(s)}, where i as 

 usual represents y^( l). From this it appears that the transformation 



where F and f are functions having real values for all real values of their arguments, 

 makes the real axis in the w plane correspond to a curve in the z plane whose intrinsic 

 equation is x = F(.s'), the relation between w and s being s=f(w), where f is 

 an arbitrary function. But transformations of this type, though they settle the 

 correspondence of prescribed boundaries, oft'er no guarantee against zeros, infinities 

 or singularities at points where w is not real, and so do not necessarily give conformal 

 representation. The geometrical idea of the transformations is nevertheless useful. 



2. Some problems of conformal transformation with partially curvilinear boundaries 

 were worked out by Mr. W. M. PAGE in a paper* published by the London 

 Mathematical Society a few years ago. Mr. PAGE definitely rejects, on account of 

 an apparent indeterminateness, the method of treating a curve as the limit of a 

 rectilineal polygon and seeking the limit of the product of the corresponding 

 Schwarzian factors. Instead he has recourse to the empirical but useful device of 

 writing down the Schwarz transformation as it would be for a rectilineal polygon 

 having the same angles as the prescribed curvilinear one, and then introducing into 

 the expression for dz/div a further factor which has the effect of making the relevant 

 side of the z polygon curvilinear while leaving the other sides straight. There is no 

 question of prescribing the form of the curve in the z plane, one simply takes such 

 conformal transformations as one can succeed in formulating in the above manner 

 and tries to find out what sorts of curve they yield. 



Mr. PAGE, acknowledging a suggestion by Mr. H. W. RICHMOND, takes factors 

 which occur in well-known problems, and by associating them with new sets of 

 Schwarzian factors obtains a new series of results. The familiar case of a semi- 

 circular boss on a straight line, namely 



(1) 



supplies the factor w+ (^-c 2 ) 1 '-', and the case of a semi-elliptic boss on a straight line, 

 namely 



}, ..... (2) 



supplies the factor w sinh a+ (w 2 -c 2 ) l/ > cosh a. These two factors and their square 

 roots ^are the factors which Mr. PAGE associates with other factors of the Schwarzian 

 type in order to obtain new results. 



" Some Two Dimensional Problems in Electrostatics and Hydrodynamics," ' Proc. Lond. Math. Soc., 

 Ser. 2, vol. XL, 1912-13, p. 313. 



