Egberts — On some Properties of a Certain Quintic Curve. 37 

 But, by the theory of partial fractions, we have 



1 6 6^ 6 ^ 



Consequently we obtain the following three relations connecting the 

 five values of 6 and their di:fferentials which correspond to the points 

 of section of a line with the curve, 



S^=0, 2^=0. . (5) 



J -Si J -^1 J ^1 



These differential equations we can integrate since they are true for 

 every line which can be drawn to meet the curve. 



If we now write C 6'' dd 



r being an integer, we obtain, by integration of the differential 



.system (5), 



^1^(0) = c,, - (6) 



^ denoting summation from 6i to 0^ , and Cq , Ci and Co being constants. 



4. — "We now proceed to determine the values of the constants 

 Co , Ci , and c^ . 



By reference to equations (4), Art. 1, we see that to a given value 

 of the parameter 6, corresponds but one point P, if we agree to affect 

 the radical ^j5 with a positive sign, its corresponding point P' being 

 determined' by giving the radical a negative sign. To the triple 

 point will then correspond three values of the parameter ^, or, in 

 other words, there are three different values of 6 which will give us 



X = 0, y = 0, 



these values of 6 corresponding to the different branches of the curve 



R.I. A. PROC, VOL. VIII., SEC. A.] D 



