SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 



1097 



When the eccentricity is increased by giving a different value to c, the difference 

 between the y coordinates of the oval and the ellipse also increases, as shown in the 

 following table, where r 02 = '9819 ; r m = '0859 ; c = '4480 ; a = '5339 ; b = '3380 ; other 

 values of r, r. 2 being as above. 



x = x 



•5339 



•4743 



•4009 



•3267 



•2577 



■1882 



1240 



0615 



00 



y 



00 



1634 



2313 



•2733 



•3007 



•3184 



•3298 



•3362 



3380 



y' 



00 



1550 



2232 



•2670 



■2967 



3162 



•3287 



3357 



•3380 



A = (y -y') 



00 



0084 



0081 



•0063 



0030 



0022 



0011 



0005 



0000 



In PL IV. fig. 6, the exterior curve of each pair represents the sextic oval, and the 

 interior curve the ellipse, as traced through the points here given. 



Further researches as to the properties of elliptic ovals may be expected to yield 

 interesting results ; and it appears to me that these curves, and the curves obtained from 

 them by linear transformation, are capable of expressing the facts of a certain class of 

 physical problems with greater accuracy than is obtainable by the tables at present in 

 use for the purpose. 



The three curves shown in Plate V. fig. 6, are linear transformations of the sym- 

 metrical oval. Their equation is 9^/\+^//* = l; \ was taken =1; and /x. was taken 

 successively = 2, \ and f to obtain the three curves. In the diagram each curve is con- 

 nected with its foci by lines drawn for the purpose. The same equation with different 

 values of either c or /x gives a different curve, as the figure shows. The complete 

 equation is 



A" \»- 



n 



M 



I+ + 



/•:; 



= A" 



and the curve may be described as the Oval of single symmetry. The Cartesian oval is 

 a limiting form of the oval of single symmetry of the 4th degree, as may be verified by 

 twice squaring its equation r x j\ + r.,jfju = A. 



The curves formed by giving negative values to one of the quantities, or to one of 

 each pair of homologous terms, are remarkable for their varied and fantastic forms ; but 

 I have not been able to discover any properties which are common to the class. 



The curve of the symmetrical equation r% — r\ = A (shown in PL V. fig. 5) is an 

 inflexional curve of two branches. Each branch crosses the axis of X, and is symmetrical 

 on either side of it ; after being inflected on either side, the branches continue to 

 approach to the asymptote Y, which accordingly has double contact with the curve at 

 the point infinity and also at negative infinity. Generally, for the symmetric radial 

 equation of any number of terms of even powers, I have found (1) in the form (j3) or 



( H h ), if the coefficients of the intermediate terms are fractional, the curve is an 



oval entirely concave to the centre ; (2) if the coefficient (in the form /3) exceed unity 



