SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1079 



(2) If all the pairs of homologous terms of the symmetrical expression have like 

 signs, but some of the pairs are positive and some are negative, the curve is the In- 

 flexional oval, for all positive values of the coefhcients ivhich are less than those of the 

 binomial expansion, and for all fractional negative values. Outside these limits the 

 curve is a hyperbolic, with alternate real and conjugate branches, the limit between 

 the closed and open forms being the continuous hyperbolic in which all the branches are 

 equal and real. 



(3) If all the pairs of homologous terms have unlike signs, and if the equation when 

 arranged in binomial form has the terms (being all even) alternately positive and 

 negative, the curve consists generally of n hyperbolic real branches, with alternating con- 

 jugate branches ; but for certain values of the coefficients, the number may be reduced 

 to two equilateral real branches, having the equilateral hyperbola as a limiting form. 



(4) If all the pairs of homologous terms have unlike signs, but if the positive and 

 negative terms do not follow in alternate order, the curve consists of two equilateral 

 inflected branches, the curve being concave to the centre and to the asymptotes where it 

 crosses the axis of X, but after inflexion on either side of that axis becoming convex 

 to the asymptotes. 



The same form, where some of the pairs of terms have like signs, and some have 

 unlike signs, except that the assymptotic axes are not rectangular. 



(5) In all cases where the equation is reducible to the two-term Polar form, 



r n cos(p0) = l , 



the curve consists of a number of alternate real and conjugate branches, which are all 

 equal. The number of such forms evidently is n/2, because p may have the series of 

 values, n, n — 2, n — 4, &c. 



(6) If the equation is not a symmetrical expression, but is homogeneous, the curves 

 fall into the above categories, but have not in general secondary axes. 



15. Determination of Contour-lines of Homogeneous Surfaces. 



If v x v 2 be coordinate quantities of any symmetrical diametral equation (suppose of the 

 form a), and if x 2 + z 2 be substituted for v\, and y 2 + z 2 for v\, and the equation be expanded 

 in terms of powers of x 2 , y 2 , z 2 , we obtain the equation of a symmetrical homogeneous 

 surface referred to conjugate diameters. The equation then takes successively the three 

 forms which follow — 



v{+~Pvtvl+~Pv*vl+vl=l (1) 



(x 2 +zJ + P(x 2 + z 2 ) 2 (y 2 + z 2 ) + 'P(x 2 + z 2 )(y 2 + z 2 ) 2 + (y 2 + z 2 y = l . . (2) 

 x 6 + Sx 4 y 2 + Sx 2 y 4 + z 6 + V{x 4 y 2 + x V + 2x 2 y V + 2x 2 z 4 + z 4 y 2 + z 6 ) 



+ V(y 4 x 2 + y 4 z 2 + 2x 2 y 2 z 2 + 2y 2 z 4 + x 2 z 4 + z e ) + y« + Sy 4 z 2 + 3y 2 z 4 + z° = 1 . . (3). 



If we suppose the equation to be given in the form (3), we can only find values of 



