APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 85 



with the necessary and sufficient condition that Q (x, y) be a poly- 

 nomial of degree m — 3 in x and y, and that the curve 



Q,{x,y)=0 



shall have as multiple points of order i — 1 the multiple points of 

 order i of the curve/* (a;, 2/)=:0'. Eeplacing the integral in (10) by 

 (11) and differentiating, Abel's specialised theorem may also be 

 written in the form 



2 ^ ^ ^ !1=0. (12) 



72,= 1 f\ {a^nyVn) 



§2. Intersection of a Cubic by a Straight Line, and Gen- 

 erally BY A Curve of the titb. Order. 



1. If in formula (12)/* [x, y)=0 represents a plane cubic, then 

 Q (x, y) is a constant and Abel's theorem becomes 



^=/* dx, 

 2 ! =0. (13) 



-.0, (14) 



Let the variable curve (3) be a straight line, then 

 dx^ dxj dx^ 



7VK^) 7\¥^ 7\¥^ 



where (a?,, y,), (x^, y^), (x.^, y^) represent the points of intersection of 

 the straight line with the cubic, and assume the equation of the cubic 

 in the form 



y^=^x'—g^x—ff,. (15) 



From analytic geometry it is known that the equation of every cubic 

 may be reduced to this form by collineation, so that by assuming 

 (15) nothing is lost in generality. 



Establish now the elliptic function jt?^ (^) ^25^3)5 ^^^^ from the 

 well known differential equation 



(') PiCARD, loc. cit.. p. 401. 



(2) Here and in the following sections where the Weierstrassian sign for elliptic functions is 

 needed, the character^ shall be used in its place. 



