of the hyperelliptic functions. 239 



(i) if 4 = 0, B = 0, then Ctf = — u~\ and <z, y are rational 

 functions of u, while w is a rational function of x, y ; 



(ii) if 4 = 0, 5 + 0, then u = ~\og(G +B(t), t = B/(e~ Bu -C), 

 and («) is a rational function of e Bu ; 



(iii) if 5 s 4=440, then (t - a)/(t - /3) = e c ^-^ u , and (/> (w) is 

 a rational function of e c ( a -/3) w ; 



(iv) if £ 2 = 44(7, then m = —l/G(t— a), and ^) (w) is a rational 

 function of £ 



The theorem is then proved. We notice that the integral u 

 may be of the first, second, or third kind ; it is in the last case 

 that the limiting places spoken of come into consideration, and 

 the theorem has then connection with Picard's theorem as to the 

 values possibly not taken by an integral function. That the 

 foregoing proof is capable of reduction to more fundamental 

 considerations of a general character is sufficiently obvious ; it 

 suggests however the need of detailed investigation of algebraic 

 constructs in several variables, and in particular the question of 

 the existence of exceptional curves distinguishing constructs not 

 expressible by Abelian functions to generalise the exceptional 

 Weierstrass points of an algebraic curve. 



16—2 



