582 PROCEEDINGS OF THE AMERICAN ACADEMY. 



2) From the point of view of the physicist it may be desirable to 

 demand in the definition more than was demanded above, namely, that 

 (1) hold for all regular 6 closed curves lying within T, not merely for all 

 circles. Since it is well known, and readily proved by an application of 

 Green's theorem, that (1) is satisfied for all such curves when u is any 

 harmonic function, it is clear that this will not affect the class of functions 

 covered by the definition. Whether included as part of the definition or 

 proved as a theorem, the fact just stated enables us to see almost intui- 

 tively that a harmonic function remains harmonic when the plane is sub- 

 jected to any con formal transformation, the reasoning being the same as 

 was used in the proof of theorem IV above. 



3) The introduction of this general idea of conformal transformation 

 enables us to extend at once our definition of harmonic functions to the 

 case in which T is a region not in the plane but on any analytic curved 

 surface. 7 The definition of a harmonic function on such a curved region 

 T would be identical with the one given above for a plane region, if we 

 understand the partial derivatives there referred to to mean the derivatives 

 along the surface, and if we require that (1) hold for all regular (or even 

 analytic) closed curves on T. If now we transform the region T, or any 

 part of it, conformally onto the plane, we see at once that the function 

 thus defined in the plane is harmonic, and hence the function on the 

 curved surface was analytic there. — Here, as before, we may if we 

 choose require that (1) be fulfilled only for certain classes of closed 

 curves. There seems, in general, no object in doing this. In the case 

 of the sphere, however, the developments given above can be carried 

 through almost word for word if we restrict our attention to circles. 



4) Finally, it may be mentioned that this method of approaching the 

 subject is particularly convenient in connection with the theory of con- 

 jugate functions. Two functions u and v are said to be conjugate in a 

 plane region T if throughout this region they are single valued, continuous, 

 and have continuous first partial derivatives and satisfy the equations 



9u 9v 9u 9v 



9x 9y y 9y 9x 



If we knew that u and v had continuous second partial derivatives, we 

 could infer at once that they are both harmonic. To establish the con- 



6 A curve is said to be regular if it is continuous, and is made up of a finite 

 number of arcs eacli of winch has a continuously turning tangent at every point. 



7 Cf. C. Neumann, Math. Ann., 10, 569 (1876). 



