180 Mr Love, On the Theory of Discontinuous [May 4, 



The relation between Q and z by which the representa- 

 tion of the XI region upon the z region could be effected is 

 unknown until the problem is solved, but, by applications of 

 Schwarz's transformation given in Prop. TL, the XI region and 

 the w region can each be represented upon the same half-plane 

 in the plane of a new variable (u). In this way we can transform 

 the X2 region into the w region and hence arises a relation 

 £l=f(w) or log(dz/dw)=f(w). This is a differential equation 

 defining z as a function of w, and when it is solved we shall 

 know the region of z which is represented upon the region of 

 w. Part of the boundary of this z region is prescribed and will 

 inevitably agree with that obtained by solution of the differential 

 equation, the determination of the other part will give the form 

 of the free stream-lines. 



It is in general most convenient to take the constant velocity 

 along the free stream-lines to be unity so that the corresponding 

 value of the real part of X2 is zero, and to change the method 

 of procedure sketched above by eliminating w and finding a 

 differential equation between z and u. The region of the z plane 

 within which the motion takes place will be that which by this 

 relation is conformably represented upon the half-plane u. 



In fact we have a given polygonal boundary (formed of parts 

 of parallel lines) in the w plane, and this can be transformed 

 into the real axis in the u plane by a relation of the form 



dw , , . 



sir ■£<">■ 



And we have a given polygonal boundary in the O plane, and 

 this can be transformed into the real axis in the u plane by a 

 relation of the same form, say 



d£l , 



where the roots, poles and critical points of f 2 (u) are points 

 u = a, ... assumed to correspond to the corners of the polygon in 

 the X2 plane. Integrating this equation we obtain 



dz_ = Ce fU{u)du 

 dw ' 



where G is a constant of integration. Hence we have the dif- 

 ferential equation 



^- = Cf 1 (u)e fMu)du . 



OjUj 



If we integrate this for the parts of the line u real that lie 

 between the critical points of the function X2, we shall find ex- 



