86 nNIVERSITY OF COLORADO STUDIES 



it is immediately seen that 



x=pu, y^=p' 21 (16) 



satisfy (15). To every value of u corresponds a point of the curve, 

 since jp a.nd p' are uniform and the point (x, y) remains the same if 

 the argument u is increased by multiples of the periods. Conversely, 

 to a point of the curve corresponds one and only one value of u. 

 The application of (16) to (14) gives, since 



dx^=p' (-?*) du, or -j-=du, 

 and from (15) s^=2y, o-=^;7— r=-o-> 



Vx Vi Vz ' 



or designating the values of u corresponding to the three points of 

 intersection by «„ -Wj, -Wj, 



du^ + du^ + 6?'M3=0, 

 or u^-\ru^-\-u^-=co7ist. (17) 



Hence: 



The sum of the arguments corresponding to the three points of in- 

 tersection of a straight line with a cuhic is constant. 



To determine the value of this constant assume the a?-axis as a 

 particular position of the variable straight line. Then p'u =0. The 

 values of u for which this is the case, are the half-periods w^ vj„ and 

 w H- w^, consequently also the values w + 2kw + 2k{U)^, w, + 2hv 

 + 2A\w^ , w-\-w^-\- 2kw + 2k^ Wj . 

 Hence, designating by I and l^ new integers, 



'f^i + '^2 + u^=^2io + 2w^ + Qk'^ + 67i:,'?z>, =^2lw-\- 2l^w„ 

 or u^-\-nj-{-n^=0 (mod. per.)'. (18) 



(1) Followins this = o shall always signify ^= o (modulus periodicity). 



