Roberts — On some Properties of a Certain Quintic Curve. 43 



9. — Given the curve, to determine the Q points. By aid of the 

 theorem of the h^st Ai-ticle, we can find the Q points by drawing any 

 line meeting the curve in five points. By means of the ruler alone 

 we can determine their five corresponding points, and through these 

 latter points and it will he found possihle to draw a conic which 

 will meet the curve in the required points. 



Since we know the Q points, we can draw the tangents at the 

 triple point. 



All we have to do is to draw a line through the Q points meeting 

 the curve in three other points, the lines joining these to will 

 touch the curve at . 



10. — If any line be drawn through one of the Q points, Q, 

 meeting the curve in four other points, their corresponding points 



lie on a line which passes through Q' . 

 "We have 



%I{6) + J= JY, 



2 referring to the four ^s of the points in which the line through Q 

 meet the curve. Xow 



J+ J' = 2iV; 

 therefore, 



N-J= J' - N; 

 consequently, 



- 2 I{&) + J' = N, 



which proves the theorem. 



On account of the importance of this theorem, we give another 

 proof. 



Let the factors of Q be $' and j' , so that 



and let us seek the equation which determines the points in which 

 a line z = Xg-, drawn through one of the Q, points meets the curve. 

 Substituting the value of s in the equation 



^z2-2^s + ^(3 = 0, 

 we find 



AX-f - 2B\q + Aqq' = 0, 



or. dividing by X j , 



A{\q + k-'q') - 2B = (1) 



