322 Transactions. 



The equation of the circle of radius HP concentric with the circum- 

 circle is 

 a cos AX + b cos BY + c cos CZ - 2 A E = 4EHP 2 sin A sin B sin C. 



H ftficft 



HP 2 (a cos AX 1 + b cos BY 1 + c cos CZ 1 - 2 a E) = 2 aB\ 



Eliminating HP 2 , the equation of the inverse of U becomes 



a cos AX + b cos BY + c cos CZ - 2 a E _ 2 A E 



IX + mY + ?iZ ~P~ 



or h* (S + 2 a E) = 2 a E (ft 2 + U) ; 



i.e., /rS - 2 a EU = o. 



The equation of the circle which is the inverse of the line ZX + mY 

 + nZi — h? = o takes the same form as the above, subject to the condition 

 I + m -+■ n = o. 



The equation of the circle which is the inverse with respect to, the 

 circle ABC of the line pa + qfi + r y = o is 



aX {p cos 2A + q cos (A — B) + r cos (C — A)} 



+ bY {p cos (A -- B) + q cos 2B + r cos (B - C)} 



+ cZi {p cos (C - A) + q cos (B - C) + r cos 2C} 



= abc (p cos A -\- q cos B + r cos C). 



The equation of the circle which is the inverse with respect to the 



circle ABC of the line — (- ^- -f 5- = o is 



a o c 



a cos (A — w) X + b cos (B — a>) Y +• c cos (C - w) Z — abc cos oo, 



where w is the Brocard angle of the triangle ABC. 



