238 Dr Baker, On the differential equations 



and from this <£' (u + u ) is expressible in a similar form. These 

 equations however are those of a birational transformation of the 

 construct into itself involving an arbitrary parameter ; we know 

 that such a transformation is impossible unless the construct is 

 of class unity or zero ; and the proof of this result is entirely of 

 an elementary character, depending on the existence in all cases 

 when the class is greater than unity of a finite number of ex- 

 ceptional places upon the construct — as for instance the inflexions 

 of a plane quartic curve (see Weierstrass, Werke, II. p. 235, and the 

 writer's Abelian Functions, pp. 44 and 653). 



When the construct is of class unity (j>(u), <f>' (u) can be 

 expressed rationally by a pair of elliptic functions, and therefore 

 by the elliptic functions |p (u), g)' (u), and conversely. In such case 

 u is expressible in a form 



dg 



-I 



and is without infinities, or the limiting places spoken of above 

 are absent. 



When the construct is of class zero, x = </> (u), y = <f>' (u) can be 

 expressed rationally by a parameter t, itself a rational function of 

 x and y ; so transformed let 



u=! R(t)dt, 



and suppose t so chosen that t = oo does not make u infinite. In 

 the neighbourhood of any finite value of t, other than the infinities 

 of the integral, we have a form 



u = A + A 1 (t — a)+ ..., 



and, as t is a single-valued function of u, the coefficient A 1 cannot 



be zero ; thus -^- is not zero for a finite value of t, other than 

 dt 



those making the integral infinite ; clearly R (t) is not zero for 



finite values of t which do make the integral infinite. Hence 



R (t) vanishes for no finite values of t, and 1/R (t) is a polynomial. 



Near % = t~ x = Q, 



--/f« e G)* 



is not infinite, by hypothesis, and as before t 2 R (t) is not zero for 

 t= oo . Thus on the whole 1/R(t) is a polynomial of the second 

 order. Putting then 



dt 

 ^A+Bt+Ct 2 ' 



