322 Transactions. 



If two circles of radii R + p and R — p be described concentric with 

 the circumcircle of the triangle ABC, the tripolar equation of the pair of 

 circles is 



a VX - p 2 + b Vy~^7 2 + c VZT^ 2 = o. 



To prove this, let any point P be taken on the circumcircle : If A -f p, 



+ v = o, the co-ordinates of P may be taken to be (_,_,_) where 



h \A jx v) 



h = - + - + - • 



A p. v 



The equation of the circle of radius p having its centre at P will be 



" 2 X + b *Y + c -Z = p>( a : 2 + b2 + c - 

 A fx v \A fi. v 



i.e., f (X - p 2 ) + b * (Y - p 2 ) + - 2 (Z - p 2 ) = o 



A 



/*■ 



V 



and the envelope of this circle as A, p., v vary is 



a VX -p 2 + b VY - p 2 + c VZ^7 2 = o (5) 



and this envelope consists of two circles of radii R + p and R — p con- 

 centric with the circumcircle. 



This equation (5) when expanded is 

 2 (a 4 X 2 ) - 22 (6 2 c 2 YZ) + 4ai>cp 2 2 (a cos A X) - 16 a 2 p 4 = o. 



Subtracting this from the fundamental relation (4) multiplied by 4R 2 

 we have 



4R 2 \ 2 (a 2 cos 2 A X 2 ) + 22 (be cos B cos C YZ } 

 - abc (2R 2 +p 2 ) 2 («cosA X) + a 2 b 2 c 2 R 2 + 4 A 2 p 4 = o 

 i.e., [ 2 (a cos A X)} 2 - 2© (2R 2 + p 2 ) 2 (a cos A X) + ® 2 (4R 4 + p 4 ) = o 

 where ® = 4R sin A sin B sin C ; 



hence { 2 (a cos AX)-® (2R 2 + p 2 ) [ 2 = 4R 2 p 2 ® 2 , 



or 2 (a cos A X) = [R 2 + (R ± p) 2 ] , 



giving a cos AX-j-i cos B Y + c cos C Z 



= 4R sin A sin B sin C [R 2 + (R + p) 2 ] , 



a result in accord with that previously given. 



In a similar manner by supposing the centre of a circle of radius p to 

 move on the line la + mfi + fly = o we obtain the equation of the pair of 

 straight lines parallel to la + mfi -\- ny = o and distant p from it in the 

 form 



- % (F 2 - 4YZ) + T% (G 2 - 4ZX) + j2 (H 2 - 4XY) 



2mn 2nl „ „ 2lm 



- -^-( GH - FX ) - ^( HF - GY ) - ^( FG - HZ) - o 



where F = Y + Z-a 2 -2p 2 



G=Z + X - b 2 - 2p 2 

 H-X + Y - c 2 -2p 2 



