Mr. Stubbs on a neta Method in Geometry. 339 



from this, that r r' — k% then r d r' = — K d r, and 



r r* rdQ r'dQ „ . ,. 



— — = j-.. or —j— = 3-r-. Hence the perpendicu- 



rfr rfr ar flr r r 



lars from the pole on the tangents are as the radii, and the first 



perpendicular being known, the second is so also. 



Having established these principles, I shall proceed to show 

 the application of this method, first to the right line and 

 circle, afterwards to curves of the second degree, and finally 

 to surfaces. 



If a circle passing through the pole be f^g. 2. 



called a polar circle ; from the known theo- 

 rem of the bisectors of the angles of a tri- 

 angle meeting in a point, by taking the 

 inverse of all these lines we come to the 

 following theorem : if three polar circles 

 form by their intersection a polar triangle 

 ABC, the polar circles A O, BO, CO 

 bisecting the angles meet in a point O, 

 which is the inverse of the point in which 

 the original bisectors meet. 



From the theorem of the three perpendiculars from the 

 angles of a triangle on the opposite sides meeting in a point, 

 we get by inversion the three polar circles perpendicular to the 

 opposite sides of the polar triangle, and passing through the 

 angles meet in a point. 



In like manner every theorem in plane geometry, com- 

 prising only the right line and circle, gives a conjugate one, 

 in which right lines and circles only are contained, — every 

 theorem, I mean, which has relation only to position, without 

 introducing lengths of lines. I shall not mention any more 

 of them, as, when the principle is clearly seen, that to a line 

 corresponds a circle, and to a point a point, to the contact of 

 a line and circle, the contact of two circles or of a line and 

 circle according as the pole is or is not on the circumference 

 of the circle, and to the angle between two lines, the angle 

 between the tangents to two circles at their point of inter- 

 section, any one can multiply theorems at will. 



The general equation of a conic section being 

 A^ + Btf^+C^ + D^ + Etf + F = 0, 

 substituting for y f r sin 0, and for x, r cos 0, we get 



Ar 2 sin 2 + Br 2 sinflcos + Cr 2 cos 2 + Br sin0 + Er cos 

 + F = 



A 2 1 



the pole being at any point, put r — —7-, or —7- for sim- 

 plicity, and 



Z2 



