

SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1093 



These equations do not presuppose any relation between A and c. Accordingly, by 

 varying the distance of F and 0, the equation may be made to represent an oval from 

 any two points on the major or minor axis taken as poles, and these poles may be either 

 interior to, or on the oval, or (within certain limits) exterior to it. If a and b represent 

 the lengths of the principal semi-axes of the oval, the limiting position of an exterior 

 pole or focus is c = 2a, where the curve is reduced to a line. The limiting positions of 

 interior poles or foci is of course c = 0, where the foci coincide, and the curve becomes a 

 circle. 



The principal foci of the oval are determined under the same conditions as the foci of 

 an ellipse ; by taking c 2 = cr — b 2 . Then, for the pair of equal radii drawn from the foci 

 to the extremity of the minor axis, we have the relation )\ = r-> = a. If the major axis be 

 taken as of unit value, then c 2 = (a 2 — 6 2 )/a 2 = e 2 , and the polar equations for the 4th and 

 6th degree curves may be written 



R 4 +2e 8 K 2 (l + 2cos 2 0) = (A 4 ~2e 4 )/2 (5); 



R 6 +(3e 2 K 4 +3e 4 R 2 Xl+4cos 2 0) = (A 6 -2e 6 )/2 . . . (6), 



where e is the eccentricity estimated in the same manner as in the case of the ellipse, the 

 cognate curve of the 2nd degree. And similarly for any curve of even degree in which 

 the foci are properly taken. It may here be noticed that the equation K 2 (l + c 2 cos 2 6) = A 2 

 represents an ellipse, because it may be immediately changed to y 2 + c 2 x 2 = A 2 . The ellipse 

 then belongs to this family of curves, of which it is of course the lowest form. 

 Plate V. fig. 4, represents the sextic curve having the equation 



^ + 15r{r 2 + 15r 2 r 4 + r!;=l, 



and referred to its principal foci. For its construction the values of rir 2 are used, which 

 are transcribed in the ensuing table, p. 1096. But as the curve was to be referred to its 

 principal foci, it was necessary to adopt as the maximum and minimum radii the pair of 

 values whose sum is equal to twice the mean radius, r 21 . Hence the only available 

 values were 



r 2 = f 7652 7115 6603 "6103 -5611 ) # 



r x =(-3569 -4108 -4623 "5122 5611 J * 



The curves here considered have a general resemblance to ellipses ; and if the equation 

 in x-smd-y coordinates be referred to oblique axes, the curve resembles an ellipse referred 

 to conjugate inclined axes. The greatest and least diameters are thus apparently conju- 

 gate, but are not really so ; because it has been found impossible by analysis to reduce 

 the locus of mid-points of parallel chords to a simple equation. It will be seen from 

 Plate V. that the difference between the biradial curve of the 6th degree and the ellipse 

 described on the same axes is very small, and it is probable that the class of homo- 



* These coordinates are very nearly the same as those of the ellipse described on the same axes. 



