SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1095 



If the equation consists of only a middle term equated to unity, or o\'.r 2 2 = 1, this 

 is immediately reducible to r\. r|= 1, which gives by transformation, 



(#2 + f + cj _ 4(^2 + yiy CQS 2 Q s 



the equation of the oval of Cassini. 



There is a curious relation between radial equations and equations of the same form 

 in rectangular coordinates, which is connected with the value of the coefficients. 



It has been seen (p. 1072), in the case of the sextic equation of positive even powers 

 of x and y, ( 1) that if P, the coefficient of the pair of intermediate terms be = 3, the curve 

 is a limiting circle ; (2) that by varying P from to 3 we obtain every form of the non- 

 inflexional oval ; and (3) that for values of P exceeding 3 and less than 15, or n(n — 1)/2, 

 we obtain the same series of curves turned round through an angle of 45°. In the case 

 of the sextic radial equation of positive even powers of i\ r 2 , if P, the coefficient of the 

 pair of intermediate terms = 3, the curve also reduces to the circle (rl + rl = 1). If P be 

 less than 3, the radial equation represents a curve referred to foci, or poles, in the line of 

 the minor axis, which may be either interior or exterior or on the curve (see figs. 1 

 and 2) ; and this includes the case of the equation of three terms, where P = 0. But, if P 

 exceeds 3, the radial equation represents a curve referred to foci, or poles, in the line of 

 the major axis, which may be either interior or exterior or on the curve; and this includes 

 the case of the curve referred to its principal foci. 



It appeared to me that the radial curves, as traced, were a little more rounded at the 

 apses than ellipses, and this impression has been confirmed by the numerical computation 

 and comparison of the forms of the ellipse and of the sextic oval described on identical 

 major and minor axes, which will be immediately given. It would be interesting to make 

 a cognate comparison for elliptic ovals of different degrees. There are two ways in which 

 such a comparison may be instituted. 



(1) If the vertices of the curves be taken for foci, or poles of radial coordinates, the 

 arbitrary term is then a parameter, and a series of curves of different degrees may be 

 described upon the same principal axes, a and b. (2) If the foci of the normal position 

 (c 2 = a 2 — 6 2 ) be taken for poles, it is difficult to prearrange the equations so that the 

 curves to be formed shall have the same amplitude. We may, however, compute each 

 curve independently, and compare it with the ellipse described on the same axes, and 

 thus find out for the curve of any degree how far its coordinates differ from those of the 

 ellipse of equal amplitude. In either case, it is necessary to reduce the radial coordinates 

 to rectangular. This is easily done. Keferring to p. 1092, we see that 



/y& /y*% 



T \ — Ti = 4eRcos0 = 4ca;; •'• x =~^ — 

 Also, 



rl+ri=2(x i +y 2 +cy, .: ,/ = *!±^ _ (^ + e 2 ) • 



