J900.]' P. Dutt — Properties of the circle and analogous matters. 95 



Now, let us consider the properties of the system of circles, 



obtained with the same fixed points A, B, by varying the ratio of AP 



GA 

 to BP. We have —-^m*. 

 LB 



AB . , - AB 



.-. — - = m*- 1 or CB 



CB m 2 -l 



Now, AB is constant ,\ CB oc — - — r. (15). 



m* - 1 



If m = l, GB—y> , and this must be so as in this case, the locus 

 reduces itself to a straight line, which may be taken as the case of a 

 circle with an infinite radius. 



It will appear from (5) that all the circles of the S3 T stem are 

 intersected orthogonally by the circle described on AB as diameter. 



(AB \ 2 

 -~s — 7 ) • 

 m % — 1/ 



m % — 1 



ovAB = r(m--\ (16) 



This shows that the radius remains the same if m be changed 



into . 



m 



Geometrically this means that if we describe a circle making 

 BP = m . AP, the radius will be the same, but the centre will be on the 

 other side, and this also appears from the consideration of symmetry. 



The relation r 1 =m?^ reduces to r-ma where a is fixed, and r and m 



r 

 vary. Writing the relation in the form — = a, we arrive at another 



result, which may be enunciated as follows : — 



If two circles be described, one on the line joining the fixed points 

 as diameter, and the other with centre B and radius equal to a ; then 

 if any of the system of circles drawn with the same fixed points A, B 

 and having the relation r x = mr 2 cut the circle described round B as 

 centre in P, and the straight line AB in D, and BE be drawn perpen- 

 dicular to AB, meeting the circle described on AB as diameter in E, 

 then the length of AP varies as the square of the co-tangent of the 

 anode DAE. 



