of the hyperelliptic functions. 237 



and hence y = ( A 2 (x, x ) ,— ) = B{x), 



\ (tUo/ Ua = o 



an algebraic function of x, so that x, y are connected by a rational 

 algebraic equation 



which we may suppose irreducible. To every finite value of u 

 belongs a definite place x = <f>(u), y — <f>'(u) of this algebraic 

 construct ; conversely to every place (x, y) belongs a value of u 

 given by 



_ f <*• »> dx 

 "J J' 



y 



save only that those definite places which are the infinities of this 

 integral are limiting points in the neighbourhood of which u 

 increases indefinitely. There is thus a finite number of definite 

 places upon the construct not actually given by the representation 

 x = <f> (u), y = (f)' (u) for any assignable finite value of u, these being 

 the infinities of the integral ; only in the case when u = oo is not 

 an essential singularity of <f) (u) can u = <x> be assigned to these 

 places and no other places. For example in the simple case 

 y = x, the two places (0, 0), (oo , oo ) of the construct are limiting 

 points. 



Consider now the algebraic function of x occurring in the 

 fundamental equation 



<£ (u + u ) = A (x, x ) ; 



putting h = <f> (0), k = (b' (0). To every ordinary place (x, y) 

 belongs a definite value of u given by 



f (*, y) dx 

 u=\ — , 



save for multiples of the periods of this integral, these periods 

 being at most two in number since cf> (u) is a single-valued mero- 

 morphic function ; as these are also periods of </> (u + v ) it follows 

 that for every two places (x, y), (x , y ), save the limiting places 

 for which u becomes infinite, a definite value can be assigned to 

 A (x, x ); and this is true however near (x, y), (x 0> y ) may be 

 to limiting places. Consider a limiting place ; near it write, as 

 usual, x = a + t m , y—b + P(t)', either the function A (x, x ) is 

 expressible by integral powers of t or not; if not, a circuit of t 

 about £ = alters the value of the function without altering the 

 place, contrary to hypothesis. 



Thus the function A (x, x ) has a definite value even at the 

 exceptional places, and is therefore a rational function of {x, y), 

 and therefore also of (x , y ). Thus we have 



<j)(u + u ) = R (x, y\ x , y Q ) = R [<f> (u), <£' (u) ; </> (u ), §' «)], 



VOL. XII. PT. III. 16 



