86 Mr. Lubbock on sotne Problems ifi Analytical Geometry. 



^x — a.y = 0, which is the equation to the line joining the 

 centre of the circle and the point a jo. 

 It is evident that since the hyperbola 



(m^ — n^) xy — m^ cty + Ji- ^ x = 



is the same for all similar ellipses, there will be one, as fig. 1., 



which touches the hyperbola. 



In this case two of the normals, as P Ng, P N4, coincide and 



become equal ; after this the ellipse diminishes, one branch of 



the hyperbola ceases to cut the ellipse, and only two normals 



can be drawn from the, point a /3 to the ellipse. 



d y 

 When the ellipse touches the hyperbola, the values of -5-=^ 



are the same in both. 



Let n'^ x^ + m''^7f = 7i'^ m'^ be the equation to this ellipse, 

 then 



{ (?«^-7i-)ar-?M««}«'-x = m''^y {{vi^—n'')y + n^ /3} 



and since — 5- = — js 

 iinr m 



/(m^— 7J^)x— ?«^a} ifx = n^y \{7r?—n^)y + w*/3} 

 and eliminating x and y between this equation and the equa- 

 tions 



{v^ — r?) xy — m^ uy + 7i^ ^ x = 



and m^y^ + n^ x^ = m^ n^ 



the equation to the evolute of the ellipse. 



And hence from any point within the evolute of an ellipse 

 four normals can be drawn from any point to the curve, from 

 any point without it only two. 



On the Defer minatio?i of the Foci. 

 Let n-x^ + rri^ y^ = m^ n^ 

 and let the the curve be referred to any other coordinates 

 xf and y' by putting 



a («' + ci') + b {y' + /3') for x 

 and a' (a/ + «') + b' {y' + /3') for y 



Ax'^ + Bx'i/ + Cy" + Dx^ + Ey' + F = 

 A = n^d-Jt m^a!\ B = 2 {7i^ab+ m-a!V\ C = 7iH^+ m^b'^ 

 D = 2 {rfa^a! + n^'ab ^' + m' a^ a + m-ab^') 

 E = 2 {n^ ¥ /3' + n^ 6^«' + m^ b'^ /3' + ?«" b^ «■') 

 F = n^a- u'^ + 71^ b' /3'- + 2)1"' a'/3' + m^ b' /S'-^ + 2 7n^- a' Uu'^'-m- n« 



Now let the equation A x'^ + Bx'y' + Cy'^ + Dx' + Ey' + F=0 



be 



