306 BROOKS — ORTHIC CURVES. [May 20, 



When At is I, the two ellipses degenerate into two segments, 



2Z =^ f -\- i~ "* ox I, — I. 



If the line pass between the branch points, and so cut the seg- 

 ment I, — 1, two triads again coincide, but in this case the three 

 points lie on a line, and we do not have the triply tangent ellipse. 



When the line $ cuts the axis of imaginaries, 



2 -j- 2 = o, 

 we have 



TTl 



and 



It follows that am / = 5, and so mt is the reflection of/ in the axis of 

 imaginaries and wV is a pure imaginary. Then, since we know that 



i = I, 2, 3, 



we see that x-^ is the reflection of a,, in the line x -\- x = Oy and 

 that x^ is on that line. It follows that the triangle x^x^x^ is 

 isosceles and that its base x^x.. is parallel to the real axis. There is 

 again an isosceles triangle when /' is real. This triangle has its 

 vertex on the axis of reals and its base perpendicular to that axis. 

 From the discriminant of the quadratic in /^V*, 



we see that /' is real when s> ± i. In other words, if the director 

 line ? cut the axis of reals, but not between the branch points, we 

 have such an isosceles triangle. 



From the above considerations, we see that if the directorline is 

 either of the axes 



.T -f- X = O, X X = O, 



then one branch of the orthic cubic must be a right line ; the re- 



