304 BROOKS— ORTHIC CURVES. [May 20, 



But at the same time, z also describes an ellipse with its foci at 

 s= -f- I and s = — i. These two ellipses are related in such a 

 way that a point z on one of them is correlated by the equation 



x^ — 3-^ = 22 



with three points on the other. Now the foci of both these ellipses 

 are independent of the particular value of ij. selected ; it follows that 

 if we assign successive values to /Jt, we shall obtain in each plane a 

 system of confocal ellipses of such a sort that the equation 



X^ TyX^^2Z 



establishes a one to one correspondence between them. In each 

 plane the origin is the centre of all the ellipses. Applying this 

 scheme to the case in hand, we see that a triad must be inscribed in 

 one. of the ellipses in the ^- plane. But the centroid of the triad is 

 the centre of the ellipse ; so the ellipse must be the circumscribed 

 ellipse of least area of that triad. We may say, then, that : 



The triads of the orthic cubic are cut out on the curve by a particu- 

 lar system of confocal ellipses, and each ellipse is the circuniscribed 

 ellipse of least area of the triad on it. 



X. 7 he Riemann Surface for an Orthic Cubic. 

 If we examine the equation 



X^ 3;tr = 2Z 



for equal roots, we find that the double points of the ^-plane are at 

 x^^A^i and at x-=z — i. These values of x correspond to the 

 branch points in the s-plane, s = -)- i and 2 = — i. 



Let us for a moment replace the s- plane by a three-sheeted 

 Riemann surface. All three sheets must hang together at infinity, 

 and two sheets at each of the branch points. Let the first and 

 second sheets be connected by a bridge along the base line from 

 -|- I to infinity, and the second and third sheets be similarly con- 

 nected by a bridge along the real axis from — i to infinity. 



Select on this surface any large ellipse with foci at the branch 

 points, and any line as a director line. Now consider the contour 

 obtained by starting from a point of this inside the ellipse, going 

 thence along the line to meet the ellipse, along an arc of the ellipse 

 to meet the line, and then along the line to the point of departure. 



