130 



Mr. J. H. Michell. 



[Jan. 16, 



V satisfies Laplace's equation. Let the intrinsic equation of the 

 curve Y^o ^ n ^ ne z plane be 



where s is the arc and h the curvature. 

 Then V satisfies the equation 



along f Q . 



The transformation from one diagram to another is thereby 

 reduced to solving a potential problem with a given condition over 

 an infinite straight line. 



If \Jr is a straight line, we have simply dV/d\Js = along it. 



By this means are given two general theorems, of which one is 

 that of Schwarz, and the second may be stated as follows : — 



The transformation 



da (z — Z r ) m r 



dz {z—x s y 



gives the potential of any number of infinitely long plane conductors, 

 all in the same plane, with parallel edges and at given potentials. 



Proceeding now to the hydrodynamical problem, x, y are the 

 co-ordinates of a point of the liquid, 0, yjr, the velocity and stream 

 functions. 



The boundaries of the diagram in the w plane are all straight, and 

 therefore Schwarz's theorem will transform to a u(JEi ]?-\-iq) plane 

 in which the boundary is q = 0. 



Now, if v is the velocity of the fluid at («?/), 



^ — dz dz' 



It follows that V is constant along part of the line q = and 

 dVjdq = along the rest. 



The transformation theorem just given enables us then to find 

 V as a function of u, and therefore as a function of w. 



The general solution of the problem appears in the form 



dz , , x -\ \f(u)du 



du 



where 0(w) and/(w) are both factorial forms. 



Several particular cases are next worked out, the results of which 

 may here be given. 



