268 ROYAL SOCIETY OF CANADA 



XXTX. The equation of the circle, 

 (1 + tan- r) (1 + tan i9 tan a + tan cf) Y^ 

 tan /3)^ = (1 + tan^ « + tan^ /9) (15 + 

 tan' 6 + tan' <f>) lends itself to proving 

 the properties of the radical axis. Sup- 

 pose C is a small circle and let a tangent 

 from P pass through T. We have 

 proven that the angles at P are each 

 right angles. Then 



cos C T = cos (' P cos P T. 



Therefore 



(' + '"": "^ = (1 +,an>P70 



(1 + tan' r) ^ 



But if T iy the point {0, </>) 



1 4- tan^ P T = , = 



1 4- tan r 



(1 -f tan' Q + tan' 0) (1 + ta n' ol + tan' j?) 

 (1 + tan 'r) (1 4- tan ^ tan a + tan ^ tan j9)' 



which is the same form as though (^, ^) were a point on the cir- 

 cumference. That is if (0, 0) is a point on the circumference 



» (1 4- tan' d + tan' (^) (1 + tan' a + tan' /?) 

 (1 4- tan ^ tan a 4- tan tan ^)' (1 + tan' r) 



Here when T is a point without, it equals 



tan* PT. 



XXX. The radical axis of two circles is the great circle passing 

 through their points of intersection. Where the circles have as char- 

 acteristic elements (a', ^' , r') and (a*, /?," O the radical axis will 

 readily be seen to be 



1 4- tan Q tan a' 4- tan ( ^ tan ^ ' ^ 

 T+ tan Q tan o^ 4- tan </> tan ^ 



-1=0. 



V 



(1 4- tan' i") {\ -h tan' a' + tan' /?') 

 (Ï tan' /) (1 4- tan' o!' 4- lan'/î") 



which is seen to be a great circle. Moreover, with this definition of 

 the radical axis it may be proven to be the locus of a point ?', which 

 moves so that T P' = T P", the lines being tangents to the two small 

 circl6s For 



T '^P'T- ( 1 -f tan' 6 4- tan' cf>) (1 4- tan' a' + tan' (3') _ 



