Hogg. — On certain Tri polar Relations. 319 



Art XLVIII. — On certain Tripolar Relations : Part II. 

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



[Read before the Philosophical Institute of Canterbury, 4th June, 1913.] 



The equation ZX + mY + nZ = k, where I + m + n =\- o, represents a 

 system of concentric circles as k varies. The trilinear co-ordinates (a /3 n y ) 

 of the centre of the system are 



r 2 A l 2 Am 2 A n~\ 



where A is the area of the triangle of reference ABC. Hence the general 

 equation of the system of concentric circles may be written 



aa X -f bfi Y + cy Z = k. 



Let (X Y Z ) be the tripolar co-ordinates of the centre O, and let p be the 

 radius of a circle of the svstem. Also let this circle meet OA in the point 

 P, and lee the angles BPC, CPA, APB be called A', B\ C respectively. 



Then AP = OA - p, BP 2 = OB 2 + p 2 - 20B p cos C 



and CP 2 = OC 2 + p 1 - 20C P cos B'. 



Hence K = aa (OA - P f + b(3 (OB 2 + P 2 - 20B p cos C) 



+ C 7o (OC 2 + P 2 - 20C P cos B') 



= aa X + b(3 Y + Cy Z + 2Ap 2 



- 2p (aa OA + bp OB cos C + cy OC cos B'). 



The coefficient of — 2p may be written 



OA . OB . OC (sin A' + sin B' cos C + sin C cos B') = O. 



Hence the equation of the circle is 

 aa (X - X ) + b(S (Y - Y ) + c yo (Z - Z ) = 2a p 2 (1) 



Since ZX + mY + nZ = k represents a system of concentric circles, 

 the minimum value of U = ZX + mY + ?iZ occurs for the circle of the 

 system which reduces to a point-circle — i.e., the centre of the system. In 

 this case 



tt _ I ^ A ) 2 |~ Z f m 2 ri 2 %nn cos A" 



2 (Z) f 



Z / m 2 n? 2mn cos A\ 

 LsIn 2 AV ¥ + 7 2 + " ~bc J 



m fn* £> 2nl cos B \ n /P m 2 2Zm cos C\~| 



+ sin 2 B \c 2 + a? + ~ ~c7i / + shTC W 2 + ^ + ' ~~^ /J 



= sTa I ^ ;m ^ ( a2 ) ~^~ °^ mn ( m + n ) + ^ 2? ^ ( w + + c ^ m G + m ) 



a 2 wm + bhil + c 2 Zm 



~x(/r 



abc (ajB y + Z>y a + C a/3 ) 

 - " 2 A - = ^-Kb 



where S = a/?y + bya + ca/3. 



Hence the equation of the circle of radius p whose circle is at (a /3 o y o ) is 

 aa X + b/3 Y + c 7o Z = 2RS + 2a p 2 (2) 



