1904.] BROOKS — ORTHIC CURVES. 305 



We can choose this path in such a way that one of the following 

 three cases must arise : 



(i) The contour passes through a branch point. 



( 2 ) The contour surrounds two branch points. 



(3) The contour surrounds no branch point. 



In case (i) we know that the cubic must have a node. In the 

 second case, by going three times around we can pass continuously 

 through every sheet of the Riemann surface and therefore through 

 every value of x. Or, thinking again of the .r- plane, we have a 

 unicursal boundary. Now it happens that the ellipse we choose 

 maps into one and not three ellipses on the ^-plane. If we imag- 

 ine this to expand indefinitely we shall have to consider the bound- 

 ary as our orthic cubic. It follows at once that : 



The orthic cubic which corresponds to a line which does not pass 

 between the braftch points is unipartite. 



If the contour includes one branch point, and therefore crosses 

 one bridge of the Riemann surface, we must go along two uncon- 

 nected curves to reach all the values of x. When these two curves 

 are spread on the .T-plane they lead at once to the conclusion that : 



The orthic cubic which corresponds to a line which passes between 

 the branch points is a bipartite curve. 



XL Triads in Special Cases. 



Let us turn our attention again to the two planes connected by 

 the relation 



x^ — ;^x = 2z 



We notice that while the ellipses in the s-plane have their foci at 

 the branch points, the foci of the corresponding system of ellipses 

 are not the double points of the .^-plane, but are the points 

 X = -{- 2 and x = — 2, each of which, with one of the double 

 points counted twice, forms a triad. 



As a rule there are two triads of the curve on each ellipse, corre- 

 sponding to the two points in which the director line cuts an ellipse 

 of the system in the 2;-plane. But unless the line go between the 

 branch points it will be tangent to one ellipse, consequently two 

 triads will coincide, and the cubic will be tangent at three places to 

 one of the ellipses of the system. No part of the cubic can be 

 inside of that ellipse. 



PROC. AMER. PHILOS. SOC. XLIII. 177. T. PRINTED SEPT. 29, 1904. 



