310 Tranmctions. 



9. We may determine the equation of the two parabolas 

 which can be drawn through ABC and P(ai/3,yi) as follows: — 

 The curve whose equation is 



v^+ Vf + V^= 



a 15 y 



is the locus of points whose axes of homology touch the circle 

 ABC, while the conic 



111 



aitt liiji yiy 



is the locus of points whose axes of homology pass through 



Jl Ji Ji 



ai ^1 yu 



Let these two curves cut in the point a'/3'y' : then 

 a /3 y 



a 15 y 



will be a tangent to the circle through — —- — 



a-i Pi yi 

 We have also 



J_ J^ 1 



aa' + ^^' + yy' ~ ° 



whence, eliminating a'(3'y', we have the equation of the two 

 tangents in the form 



V'aai(/3i/3 — yiy)+ ^/ 6/3i (yiy - aja) -f- y Cyi(aia - A/^) = 



and the equation of the pair of parabolas is 



vKH)w<i:")w^7(r-i)=» 



10. Let there be four concyclic points A, B, C, D, and let 

 the position of the point D be determined by the intersection 

 of the circle ABC and the conic 



l/^y-^niya + na/S = 



Then the two parabolas through the four points will be the 

 inverses of the two tangents to the circle ABC which are 

 parallel to the line la + m(3-i-7iy = o. 



Consider the conic whose equation is 

 mc — nh na — lc lb — ma 



