92 THE ROYAL SOCIETY OF CANADA 



and denominators depend on n. Furthermore when | u | =Ro these 



values coincide with $ (u), ^ (u). 



While retaining the inequality | u | < R, it might appear that 



by varying the choice of n, the different expressions in u might differ 



in value for certain positions of u, and thus to such a position ot u 



would correspond more than one place (x, y). 



Suppose that one choice of n leads to x =^H.) and another to lllïL 



g(u) _ gi(u) ' 



where the f 's and g's aie power series in u. Let the series f, g converge 

 for I u I =R and the series fj, g^, for | u | =R^ For sufficiently 

 small values of | u | the two expressions for x are equal, since they 

 are both equal to $ (u). Hence for such values f(u)/g(u) = f i(u)/gi(u) 

 and f (u) gj (u) = f j (u) g (u). But the two sides of this latter equation 

 cannot agree for all values in the immediate neighbourhood of the origin 

 without being equal for all values in their common domain, and there- 

 fore also for all values of u contained in all the foui domains of con- 

 veyance of the f's and g's. In this latter complex of values we have 

 therefore f (u) gj (u) =f j (u) g (u) and therefore also f (u)/g (u) = 

 f 1 (u)/gi (u) excluding such special values of u as make f, g (or f j, g J 

 simultaneously vanish. Similar considerations apply to y. 



This proves that x, y are one-valued functions of u for every finite 

 value of u. It would seem a small step to proceed by the use of limits 

 to the theorem that x, y are expressible as quotients of transcendental 

 integTal functions of u. But it is not necessary to do this, for in the 

 Weierstrassian theory of functions of one variable it is known that 

 every function of u that is meromorphic throughout the finite part 

 of the plane is expressible in this way. 



The fundamental character of this method was emphasized by 

 Weierstrass when he pointed out that the existence of such expressions 

 for X, y can be established in all rigour by the mere consideration of 

 the elliptic differential equation; and that one can start directly from 

 this equation, without assuming the knowledge of any other property 

 of the elliptic functions, and arrive at the representation of these 

 numerator and denominator transcendents, merely by the use of 

 general principles of expansion. (Weierstrass, Théorie der Abelschen 

 Functionen, Crelle vol. 52, p. 62). 



SYSTEMS OF PERIODS. 



We shall consider onlj'' functions of one variable, these functions 

 can however be not merely one-valued or finitely many-valued but 

 infinitely many-valued. We shall use m for numbers when we wish to 

 imply that they are simply known to be real; if we know further that 

 the number is rational we shall use n; and finally if we know that the 



