1096 



HON. LORD M'LAREN ON SYSTEMS OF 



To determine a and b, and c (the distance between the foci), we may denote by 

 ?■(,,.' r 01 the greatest and least radial coordinates, being those which are drawn to the 

 extremities of the axis of X; and by r 2)1 the radii of equal length, being those which are 

 drawn to the extremities of the axis of Y. Then, 



For any interior foci, 



,, _ ' 02 *01 ,, _ r 2 "r r Ql 



For exterior foci, 



rp _1_ fy (Y* ry 



— 02 ' ' 01 . n _ '02 M)l 



C = 



For foci on the curve, . . . 



• ^oi = °; c = a=-&; 



For foci in the normal position r 02 +r 01 = 2r 21 = 2«; and r 21 — r 01 =c; 



In all cases, 



b= V r 2i 2 — ' 



Values of a, b, c, x and y being thus found for the elliptic oval of any degree from its 

 radial coordinates r x i\, the comparison with the ellipses described on the same axes is 

 made by taking identical values of x (or x = x'), and thence computing the relative 

 values of 



'->-£ 



.r- . 



Such a comparison has been made for the curve r 6 + I5r\r'i+ I5rl'>i + r. 2 = 1, and the 

 ellipse having the same axes, a and b ; and the values of y and y' to the argument x = x' ', 

 together with the difference (A =y — y') are given in the subjoined table. Exact values 

 of i\r., were found by the homogeneous method, whence exact values of x, y, and y' were 

 found by the preceding formulae. Taking the highest and lowest computed values of 

 r. 2 i\ for the axial radii, and with interior foci, we have J" 02 = '9367 ; r m = '1651 ; c = '3858 ; 

 a= '5509 ; b = '4075. From these elements and the tabular values of r., and r u the cor- 

 responding values of x, y, and y' are as in the annexed table. 



Arguments, < 



•9367 

 1651 



•8798 

 •2358 



•8215 

 •2990 



•7652 

 •3569 



•7115 



•4108 



•6603 



•4623 



•6103 

 •5122 



5611 

 •561 1 



x = x 



•5509 



•4656 



•3794 



•2970 



•2187 



•1440 



0715 



o-o 



Sextic Oval, y 



00 



•2218 



•2986 



•3455 



•3754 



•3941 



•4043 



•4075 



Ellipse, y 



00 



•2178 



•2954 



•3432 



•3741 



•3932 



•4040 



•4075 



A =(</-?/) 



00 



•0040 



0032 



•0023 



0013 



0009 



•0002 



o-o 



