( m ) 



or, vt'hon homngonoous variables ƒ, y, r aro Iiitroilunodj by 



J{u\ ■>/. s) (lx 



oiho = 



F 3/- 



9.'/ 



where J {x, y, :) donotos the .Taenbian of f ami of tlie pencil of tnc 

 curves R. 



Meanwhile it is clear that when a reducible integral W presents 

 itself the curves Tt are not yet uniquely determined. The lower 

 limit of the integral TT' is still ai'bitrary and what is said of the 

 squares of thetas with the argumeut W is also applicable to the 

 squares of thetas with the argument W + a. Hence the rmictions II 

 can be rejjlaced by 



I = ~p{ W + a) -e, ,f-= p{ W + a) - f„ , I' = Ji W + a) - f, , 



'>.!. ■>4 ^.1. 



which functions can be expressed rationally in Ri, R^, R3, Ri and 

 F. Evidently (he constant « may l)c regulated in such a way that 

 one of the curves 5, e.g. S^, touches ƒ in a given point i"', //, 

 and by the prescription of this point the remaining r — 1 points of 

 contact of «?! are completody determined. Thus we inter that tlic 

 existence of an elliptic integral W implies an involutory grouping of 

 the points of the curve f in such a way, that the r points of any 

 group may be regarded as the points of contact of some curve <?. 



The fact that the system of the curves R depends on an arbitrary 

 parameter is important when we consider hyperelliptic curves. For 

 then in the equation of the curve ƒ one of the coordinates, say ;/, 

 occuis only in the second power and the lational functions contain 

 no power of // high(n' than the first. So it is always possible to 

 choose the constant « in such a manner that in the i-atio Sj : S,^^ 

 and then also in the two others «^2 '■ ^-i and S^ : S^^ the term con- 

 taining If is wanting. In other words: in the case of an hyperellipiic 

 curve admitting a reducible integral we can suppose beforehand that 

 each of the curves R, by means of which tlie reducti(ni has be(^n 

 effected, breaks up into a group of r light lines, drawn through 

 the multiple point of the curve. 



In the preceding the existence of the funtions R proved to be a 

 necessary consequence of the reducibility of one of the integrals of 

 the first kind; conversely, if the existence of the functions 72 is esta- 

 blished at least one of the integrals is reducible. 



