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



<£ and a stream-function i/r, so that <£ + tyjr or w is a function of #. 

 The object of the theory is to show how in any given problem the 

 functional relation between w and z can be discovered. For this 

 purpose we consider two functions £ and fl such that 



£= dz/dw and II = log f, 



and it is well known that O is a complex quantity, whose real part 

 is the logarithm of the reciprocal of the velocity of the fluid at the 

 point z, and whose imaginary part is the angle which the direction 

 of this velocity makes with the real axis in the z plane. In 

 the kind of problems to which the method is applicable the region 

 of the z plane within which the motion takes place is bounded 

 partly by fixed straight lines and partly by free stream-lines. 

 Along the fixed boundaries the direction of the velocity is given so 

 that the corresponding parts of the boundary in the fl plane are 

 lines parallel to the real axis. Along the free stream-lines the 

 velocity is a given constant/so that the corresponding parts of the 

 boundary in the 12 plane are parts of a straight line parallel to the 

 imaginary axis. Hence the boundary in the fl plane is a polygon 

 which we know how to draw. In like manner the boundary in the 

 w plane, consisting of parts of straight lines parallel to the real 

 axis {-^r = const.,) is a polygon which we know how to draw. If now 

 we take an auxiliary complex variable u the polygons in the 12 

 and w planes can be conformably represented on the half-plane for 

 which the imaginary part of u is positive. The required trans- 

 formations are given by the theory of Schwarz and Christoffel, and 

 we are thus in a position to write down two relations 



dw j. , N 

 dO, „ , . 



from which ^ = Gf^u) e$ f * ^ du . 



The roots and poles of the function f t (u) are values arbitrarily 

 assumed to correspond to the corners of the polygon in the w 

 plane, and the roots, poles and critical points of the function f 2 (u) 

 are in like manner values arbitrarily assumed to correspond to the 

 corners of tbe polygon in the O plane. Of these values three may 

 be arbitrarily fixed, then the rest can be determined. The deter- 

 mination is made by integrating the equation connecting z and u. 

 If the limits of integration correspond to two critical points of the 

 function 12 which lie on the part of the u line that corresponds to 

 a fixed boundary we shall obtain an expression in terms of our 

 assumed arbitrary constants for one of the dimensions of the fixed 



