of the hyperelliptic functions. 235 



Addendum. 



We give now a sketch of a proof of the theorem that an 

 analytical function of one variable possessing an algebraic addition 

 theorem is an elliptic function or a particular case of such. The 

 ideas employed have, of course, no novelty*. 



It is supposed here that there is a finite range of values for u 

 within which the function (f> (u) is representable by a succession of 

 power series which are continuations of one another, so that the 

 function is monogenic ; and similarly for cf) (u ) and <f> (u + u ), the 

 range within which <p (u + m ) is so representable being a conse- 

 quence of the ranges for a and u Q . And that for the values of 

 (it), cf) (w ), (f> (u + u Q ) so obtainable for a given pair of values of 

 u and i* there exists always the same algebraic relation 



<f> (u + u ) = A [(j> (u), <f) (m )], 

 provided a proper signification be given to ambiguous algebraic 

 signs occurring on the right. 



Putting here x = <f> (u ) we deduce thence an equation 



(f) m (U + Mo) + H, [(f) (m), Wo] V 11 - 1 (U + Wo) + • • • 



+ H m [(f) (m), x ] = 0, 

 which is rational in x , so that each coefficient is of the form 



H[4>(u), ^- f^tMu'" . 

 x %(u) + % k ^(w) + . . . 



wherein f (u), f(u), ..., g (u), ... are power series in u defining by 

 their continuation within the w-region above spoken of a series of 

 monogenic functions. 



If now every one of these functions f, f,..., g , ffi,--- be 

 continued as far as possible, any value of u in the neighbourhood 

 of which the continuation circles of one at least of these functions 

 have zero for lower limit may be a singular point of the coefficient 

 H [cf) (u), w ] under consideration. In a range for u, within which 

 no such singular point arises for any one of the coefficients H, 

 the function (f) (u + u ) satisfying the equation can be continued, 

 as a function of u, and will behave like an algebraic function of u. 

 In this range too every root of the equation has the same property. 

 But a value of u, which is a singular point of one at least of the 

 functions f, f, ..., g Q , g 1} ... in one at least of the coefficients H, 

 is independent of cc fl ; if it is a singularity of one of the coefficients 

 H, it is a singularity of one of the roots of the equation ; say such 

 a value is u — c ; then w = c + u is a singularity of one root (f> (w) 

 of the equation ; this is so for the definite value c for arbitrary 



* Cf. Schwarz, Formeln u. Lehrsatze, §§ 1 — 3 ; Phragmen, Acta Math. vii. pp. 

 33 — 42; Forsyth, Theory of Functions, pp. 326 — 350; Painleve, Acta Math. xxvn. 

 p. 1. 



