346. • Transactions 



Art. XLV. — On certain Tripolar Relations : Part I. 

 By E. G. Hogg, M.A., F.R.A.S., Christ's College, Christchurch. 



[Read before the Philosophical Instikite of Canterbury, 4th December, 1912.'] 



§ 1. If any point P be taken in the plane of the triangle ABC, its 

 tripolar co-ordinates are AP^ BP^, CP^ Writing X, Y, Z for these 

 quantities we have 



X sinU = /3' + y' + Wy cos A 



Y sin^B = y^ + a? + 2ya cos B 



Z sin^C = a^ + ^^ + 2a/3 cos C, 

 where (a, yS, y) are the trilinear co-ordinates of the point P. 



The fundamental relation — due to Cavlev — connecting the mutual 

 distances of the points A, B, C, P is a^X^ + ^"^Y^ + &7? - 2bc cos A YZ 

 - 2ca cos B ZX - 'lah cos C XY - 2a6c (a cos A X + 6 cos B Y + 

 c cos C Z) -f- a'^i^c^ = O ... ... ... ... ... (i). 



If X : Y : Z = A. : ju, : V, then there are two points P, Q which satisfy 

 this relation — viz., the common points of the coaxial system of circles 



X ^ Y ^ Z 



This system of circles is cut orthogonally by the circle ABC, and 

 P, Q are inverse points with respect to the circle ABC. 



§ 2. The equation ^X + mY + nZ = « represents in general a system 

 of concentric circles as k varies. Transforming to trilinear co-ordinates 

 the tripolar equation ZX + mY + nZ = o, we have 



(a sin A + /3 sin B + y sin C) ] ^-^ ( ^^ + -J^\ 

 ^ Ism xA.\sin^B ' sm^'C/ 



sinB\sm^C sin"''A/ sin C \sm-A sin^B 



= l + m + n ,n gj^^ A + ya sin B + a/3 sin C). 



sin A sm B sm C 



Hence if I -\- m -{- n = o, the equation IX + mY + nZ = o reduces to 

 the product of the line at infinity, and the line 



a (m , n\ , B /n , l\ , y(l , m\ 



which, being satisfied by (cos A, cos B, cos C), is a diameter of the 

 circle ABC. 



The tripolar equation of the diameter through the point {a^fiiy^ is 



[{y - c") aai + a^ (6/3i - cy,)] X -f [(c* - a^) bft^ + b^ (cyi - aa^)] Y 

 + [{a^ - &^) Cyi + C {aa, - h^^)] Z = o. 



