104] NEWTONIAN POTENTIAL FUNCTION. 199 



which is the condition, 42, that the complex variable u 1 + iv' 

 is an analytic function of u + iv. Thus from the solution of 

 one problem for the surface q s may be deduced the solution of 

 any number of others for the same surface. 



If now the quantities u, v be taken as rectangular coordinates 

 in a plane, the arc of any curve is expressed in the form, 



do- 2 = du* + dv z . 



To any point u, v in the plane corresponds a point with the 

 same values of u, v, on the surface q s . In virtue of the relation 



da 2 = Mds 2 



between corresponding arcs on the plane and on the surface, 

 we see, as in 43, that corresponding infinitesimal triangles 

 are similar, or the surface q s is conformally represented upon the 

 plane. If the Z7F-plane is conformally transformed to another 

 plane XY, we have seen that we have u + iv an analytic function 

 of the complex variable x + iy and the real functions u, v are 

 potential and flux-functions in the JTF-plane. 



As we have just proved that they retain this property on 

 the surface q 3 , we see that the method of the functions of a 

 complex variable will give us the solution of any number of cases 

 upon a surface, and that the surface may be conformally repre- 

 sented on the plane in an infinite number of ways. Such a 

 representation of a surface on a plane constitutes a map. 

 Surfaces which may be conformally represented on a plane 

 may be conformally represented on each other. The theory of such 

 transformations is the subject of an important memoir by Gauss*. 

 The method here given is due to Beltrami^, and may be applied 

 even when the coordinates q lt q 2 are not orthogonal. The method 

 is particularly applicable to the case of electrical currents flowing 

 in thin conducting surfaces, and the conformal transformations 

 may be found by experiment. A thin space bounded by two 

 surfaces q s in which is distributed a solenoidal vector which may 

 be represented by a potential or by a flux-function as here de- 

 scribed, is termed a vector-sheet. 



* Gauss, " Allgemeine Auflosung der Aufgabe die Theile einer gegebenen Flache 

 auf einer anderen gegebenen Flache so abzubilden dass die Abbildung dem 

 Abgebildeten in den kleinsten Theilen ahnlich wird. " WerJce, Bd. iv., p. 189. 



+ Beltrami, " Delle variabili complesse sopra una superficie qualunque. " Annali 

 di Matematica, ser. 2, t. i., p. 329. 



