308 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



Tho centre of the director-circle is seen from (59) to be the point 



('fh — bg hg — af\ 



wliich coincides with the centre of the conic ; and, if R be the radius, 

 we have 



^ CA_-lM! + (MH^/)! _ c(a + &)-(/H^^) 

 {ah - 7i2)3 "^ {ab - Ifiy ab - Ifi 



■" (a6-/i2)2' 

 which shews that in rectangular axes the sqnare of the radius of the 

 director-circle is equal to the sum of the squares of the semi-axes of the 

 conic given in equation (53). 



That the same propositions hold for oblique coordinates may easily 

 be shewn, viz.^ the equation of the tangents to the conic from (x\ y') 

 being 



(ax^ + 21ixy 4- by^ -f 2gx + 2fy + c) X 



(ax'^ + 2hxy + hy'^-\- 2gx' -f 2fy' -|- c) 



= \{ax-^liy'^-g)x^-{lix'^by-^f)y-^gx^-fy'-\-c\^ , 



the condition that these lines may include a right angle, gives for the 

 locus of {x, y') the circle 



{ab - li^) {x^ ■\-y'^ + 2xy cos <o) 



+ 2 I (^& -fh) + {fa - gh) cos (o I a; 



+ 2|(/a-sr70 + (^6-//0 costoj^/ 



-t-c(a + &) - (/2 + ^2)+2(/^-c70 cos 0) = 

 Comparing this with the standard form 



{x-ay + 2{x - a){y-P) cos w-f- {y - /5)2 = r^, 

 or {x'^ + y^-\-2ccy cos w) — 2(a + ;8 cos w)^-— 2(/3 + a cos w)?/ 



-}-a2 + ^2 + 2a/3coso)-r2 = 0, 

 we have at once 



fh-bg Q_ hg-af 

 "'~ab-h^' '^~ab-h^' 

 which give the same coordinates of centre as before, while we have for 



the radius 



r2=a2 + 2a/3 cos (0 + ^2 



__ c{a-\-b ) - (/^ + ^g) + 2(/^ - ch) cos CO 

 ab - Ifi 



= [{fh-bgy-\-{hg - afY - {ab - Ifi) | c{a-Vb) - {f^^ + g^) 



+ -2 [ {fh - bg) {hg - af) - {fg - ch) {ab - Ifi) | cos o,J ~ {ab - P) 



