SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1063 



axes of reference, consists also of even powers of the variables. In this type of curve, 

 when referred to either pair of conjugate axes, all the terms of the variables are positive. 

 It will presently be seen that where some of the homologous pairs of terms are positive 

 and some are negative, the curve may be an inflexional oval of double symmetry, 

 passing into an inflexional hyperbolic for certain values of the coefficients. 



(2) Hyperbolic Forms. — If the axes consist of a pair of finite diameters and a pair of 

 asymptotes, the curve consists of two or more infinite branches symmetrically placed, 

 which may be either all equal or of two sets. These may be termed continuous or 

 discontinuous hyperbolics, according as the branches are all real, or consist of real 

 and imaginary (or conjugate) branches in alternate order. If in each pair of homologous 

 terms the signs are unlike, the branches are entirely convex to the centre, the inflexional 

 hyperbolic being represented by an equation of pairs of homologous terms, some of 

 which pairs are positive and some negative, or of unlike signs. 



All equiaxial curves, whether of the first or the second class, complete their 

 phases within a quadrant. In curves of the second class, the secondary axes, although 

 asymptotic, are true diameters, because the form of the equation shows that each asymptote 

 bisects all ordinates drawn parallel to the other ; that is, it bisects the intercepts made by 

 two adjacent branches, which may be either both real, or one of them real and the other 

 conjugate. 



(3) Projections of Equiaxial Forms. — By writing xja for x and y/b for y in any 

 homogeneous equiaxial equation, the equation of the projection of the equiaxial curve is 

 formed. The curves of the series which may be formed by projection have the same 

 general resemblance to the primitive equiaxial forms that ellipses and common hyperbolas 

 have to the primitive forms of the circle and the equilateral hyperbola. 



(4) Heterogeneous Central Equations. — Every equation of even degree, and contain- 

 ing only even powers of the variables (although not homogeneous), represents a central 

 curve ; and if the equation be a symmetrical expression, the curve is equiaxial. I shall 

 here only consider those heterogeneous central forms which represent sections of the 

 symmetrical homogeneous surface equation. 



It has been pointed out that every heterogeneous central expression represents a 

 section of a homogeneous and central surface taken parallel to a principal plane. There 

 is then no specific difference between homogeneous and heterogeneous central curves 

 pertaining to the same surface. The highest homogeneous part of the equation is the 

 limiting equatorial section, where the terms compounded with Z disappear, and the 

 general form of the curve depends solely on the highest homogeneous part of its 

 equation. 



(5) With regard to those curves whose equations are not symmetrical functions of 

 x and y, or xja and y/b, it is in general not possible to find secondary axes to these. 

 But the curves of unsymmetrical expression are assimilated to those whose equations 

 are symmetrical by the Kule of Signs, which will presently be deduced, and their traces, 

 computed by the homogeneous method, prove that they follow the same classification. 



