PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 309 



tions upon z is adequate to determine the corresponding values of 

 w; 1 then the representation on one plane of w, corresponding to 

 an arbitrary series of values of z upon another, is known as its 

 conformal representation. 2 



Consider such a representation of a simple function, as for 



example w = z» (35) 



which with (34) gives 



Pw = ^) 6 w = n0 (35a). 



If the point z move continuously in its own plane, so that its path 

 be any given figure, the conformal representation in the w plane 

 is completely defined by this last equation. Such representation 

 may be regarded as a very general 3 complex method of continuously 

 generating geometrical figures, the function 



f(w n , z m ) = (36) 



defining the type of generative movement, which movement how- 

 ever does not acquire a determinate character until the figure 

 represented by z is also specified. 



34. Riemann surfaces. — In equation (36) denoting an irreducible 

 algebraic equation, there will, for every value of z, be n values 4 of 

 w, such a function is said to be many-valued, multiform, or poly- 

 tropic. 5 A Riemann surface, is a surface such that the n-valued 

 function can be schematically represented thereupon as a single- 

 valued function. 6 In an expression, such as w — say ± Vz, we 



1 Such a function is called monogenic; the term was first used by 

 Cauchy, who shewed by w = f (a? - iy) cannot be regarded as monogenic: 

 cf. Grundlagen fur eine allgemeine Theorie der Functionen einer veran- 

 derlichen complexen Grosse. Eiemann, Ges. Werke. 



2 Conforme Abbildung. Gauss, Werke, Bd. iv., p. 262. 



3 Weierstrass however has shewn that monogenic functionality is not 

 coextensive with arithmetical operations. Abhandl. aus der Functionen- 

 lehre, Ber. Akad. 188], p. 90. 



* See Puiseux's memoirs. Lionville, 1° Ser. t. xv. pp. 365, 480, 1850 ; 

 t. xvi. pp. 228 - 240, 1851. 



5 If for a single value of z, w has only one value, independently of the 

 way z acquired its value, the function is said to be uniform, monotropic, or 

 single-valued: if it has in any way more than one value, multiform, poly- 

 tropic, or many-valued — (eindeutig, mehrdeutig). A monogenic, uniform, 

 and continuous function is said to be meromorphic, or holomorphic, or 

 synectic over any limited region in which it possesses the indicated charac- 

 teristics. 



6 An artifice that greatly assists the study of functions. 



