236 J)r Baker, On the differential equations 



values of u in a certain range — an impossible thing for a function 

 (j) (w) otherwise analytical, unless c be infinite. Thus the only 

 singularities of the coefficients H, which arise by singularities of 

 the subsidiary functions f , fi,..., g , ffi,---, are for u = <x>. 

 Singularities of these coefficients H are possible which are not 

 singularities of any one of the subsidiary functions f , f 1> ..., 

 g , gi,...", but such are necessarily poles. 



Thus (u + u ) satisfies an algebraic equation, whose coefficients 

 are rational in x = (f> (u Q ), and have no finite singularities other 

 than poles ; in particular there is no point such that a circuit of u 

 round it changes the value of a coefficient, namely, these coefficients 

 are single-valued. 



Hence it follows that to a given value of u belongs only a finite 

 number of values of (/> (u). For putting u = 0, the previously 

 discussed equation gives an algebraic equation satisfied by <fi (u), 

 with single-valued coefficients having essential singularities only 

 for u = oo . Let 



cf> n O) + K, (u) 4> n ~ l (u)+...+K n (u) = 



be the equation of lowest order of this character satisfied by <f> (u) ; 

 we assume that all roots of this equation are obtainable by continu- 

 ation from one root, so that the algebraic addition theorem applies 

 to all ; let K (u) be a rational symmetric function of the roots of 

 this equation, which is therefore also a single-valued function with 

 essential singularity only for u= go . Then K(u + u ) is the same 

 rational symmetric function of functions cf> (u + u Q ), each of which 

 is algebraically expressible by x = cf> (u), oc = cf> (u ) ; thus K (u -f u ) 

 is an algebraic function of a; and x . Between this function then 

 and (j) {u + u ) = A (x, x ) the quantity x Q can be eliminated. There 

 is therefore an algebraic equation 



B [(f) (u + u ), K (u + u )] = 



wherein, since u + u Q can be kept unaltered when u alters by 

 suitably adjusting the variation of u , the coefficients are in- 

 dependent of %i. In other words, putting u now zero, the function 

 § (u) is an algebraic function of a single-valued function with no 

 essential singularity save for u = oo . 



We have thus to show that such a single-valued meromorphic 

 function which has an algebraic addition equation is an elliptic 

 function or a particular case of such. 



Putting x = <j) (u), y = cf)' (u) = -j- , 



the equation <j>(u + u ) = A (x, x ) 



gives 4>' (m + Wtf) = A 2 (x, x ) -=-?■ , 



