1904.] BROOKS— ORTHIC CURVES. 325 



all the antipoints obtained by pairing these in all possible ways 

 satisfy the equation for all values of /. Now we know that the 

 {n — i)^ points thus found form an orthocentric set. We are now 

 in a position to state the following general theorem : 



If 2n — I points of an orthocentric set of iH-- points lie on a circle, 

 then the remaining {n — \f points of the figure form a central ortho- 

 centric set of which n. — i points are real. 



The vectors of the n — i real points are the roots of 



VII. The Pencil Determined dy Any Ttao Orthic Curves. 



We are now ready to consider the most general pencil of orthic 

 curves. Form the equation 



x^ — (^1 + ta!^ x^-' + {a^ -f ta'.^ x^''' — + . . . 



where / is a parameter which has the absolute value unity. Now 

 for every value of / this represents a real orthic curve of the n^^ order, 

 provided 



a^ + ta\ = (a.,^_^ + ta^n r)l \ 

 or 



I (ty ^ 2n-v 1=1 « V ^2n-v 1 • 



For if this holds, the equation can be put in the known form 

 {x — ai)(.r — a,) . . . =t^{x — a,) (x — a,) . . . (.v — a^). 

 Now let 



and 



x"" — aVT°-' + a'.^x''-'' [-..._ a'2„_i.v°-^ -f «',^.?= o, 



be the equations of any two real orthic curves. Then 



«2n = A, «2n =^2, 



and 



«v = «2n-vA \ «'y=a'2n-v4 ^' 



