30 bulletin of the 



12th Meeting. October T, 1871. 



The President in the Chair. 

 Prof. A. Hall read a paper, illustrated by a diagram, 



ON A CURVE OF THE FOURTH DEGREE. 



(^Tliis paper is published in the Educational Times, vol. xix.) 



(abstkact.) 



The curve to be considered arises from the solution of the fol- 

 lowing question, proposed in one of the English annuals: 

 "Through the focus of an ellipse a right line is drawn cutting 

 the ellipse in the points D and e, and at the middle point of d e 

 an indefinite right line is drawn perpendicular to D e. It is re- 

 quired to find the form and area of the curve that this perpen- 

 dicular always touches." 



Taking the centre of the ellipse as the origin of coordinates, 

 and its principal axes as the axes of reference, the equation of 

 the perpendicular is 



a~y 7n^ -{- (a'x — c'') m^ -{- b^y m -\- b^x = o ; 



where a and h are the semi-axes of the ellipse, m the tangent 

 of the angle which the line d e makes with the axis of x, and 

 c = ae; denoting by e the eccentricity of the ellipse. The equa- 

 tion of the curve sought will be found by eliminating m between 

 the preceding equation and its first derivative with respect to m. 

 The result of such an elimination was called by the older mathe- 

 maticians the renultant of the two equations, and in the phrase 

 of modern algebra it is called the eliminant. Performing the 

 elimination the equation of the curve is found to be — 



4 a^iy 4- (8 a^x'^ -\- 20 a'^c^x — c^) bY + 4 x (aZ^ — c3)3 = o. (1 ) 

 Putting h = a e^, and solving for y we find, 



y==±f^(yj±s/jrrrx)^{^s/T^VT:p8x)f (2); 



where the upper and lower signs in the radical expressions must 

 be taken together. 



This elegant form for the value of y follows from this fact, 

 that in the solution of (1) the terms in a: retain a cubic form 

 through two successive reductions. Equation (2) shows the form 



of the curve. It is confined to the limits ar = -j- h, and x = — - ; 



o 



and has double points for these values of x. If we denote the 

 area of the curve by A, we have 



•^-A = r' U/h^^h^Sx)' (^?j\/h—V h-\-8x)'dx 

 "- ¥^ 



—J (v/A__iv//, 4.8 a:)- (^d>/h + \/h-{-8x) dx. 



