1904.] BROOKS — ORTHIC CURVES. 315 



The path of a point which moves so that its orientation from, n fixed 

 points is constant is an orthic curve of order n. 



If a^, a2, . . . a^ are the fixed points, the condition on x is 

 expressed by the relations 



{x — ai) {x — a^) . . . {x — a„) = px^ 



and 



{x — fli) {x — a.^) . . . {x — fln) =pr~^\ 

 These lead to the equation of the curve, 



jt:n_^^^n-l_|_^^^n-2 . . . _|_ri2 (7^ . . . _^ ;~a-l _-!!■) ^O^ 



where the j-'s are the elementary symmetric combinations of the a's' 

 This is the general equation of an orthic curve. If we take x = V^^ 

 for a new origin, and make r^ real, the equation becomes 



jc° + a^x""-^ — a^x""-^ ... — a^x""-^ + a.x'"^ -f jc" = o. 



The asymptotes are the n equally inclined lines given by the factors 

 of the highest terms, 



x"" -f .T° = o. 



These lines all pass through the origin ; it follows that the centroid 

 of the n points «i, . . . a^, is the centre of the curve. Since every 

 orthic curve can be brought to the above form, we see that every 

 orthic curve is equilateral. The converse proposition, every equi- 

 lateral is orthic, is not true. The general equation of an equilateral 

 may be put in the form 



x"" + ^^° + ^ {OCX) = o, 



where (xx) is a perfectly general function of degree n — 2. <l^ con- 

 tains ^{n — 2) (n — 3) product terms, which must vanish for the 

 curve to be orthic. To make an equilateral curve orthic is, there- 

 fore, equivalent to making it satisfy ^(n — 2) (n — 3) linear con- 

 ditions. For n = 2 and n=^^ this number is zero, so the equi- 

 lateral conic and cubic are orthic. For the quartic, this says that 

 to be orthic is one condition. 



III. N-ads, J^ocif Intersections with a Circle. 

 The relation 



[x — ai) {x — a;) . . . (.V — a„) = pT^= z 



