SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1089 



direct relation that I can discover to the foci. It is certainly not the mean point between 

 the foci in the curve of single symmetry, because then the radial equation would be a 

 symmetrical expression with equal coefficients, which is of course not the case in curves 

 of single symmetry. 



It is evident that the sextic hyperboloid of one sheet, and indeed a similarly con- 

 structed surface of any degree, will furnish either oval or asymptotic curves of the quasi- 

 Cartesian type, according to the angle at which the section is taken. 



If the equation of such a surface contains only the highest power of one of the positive 

 terms, x, then when x n = the arbitrary term, the equation reduces to a pair of right lines 

 or rules. But the number of such rules apparently cannot exceed that of the conjugate 

 diameters for a hyperboloid of any even degree above the second. 



(196). Examples of Curves of Single Symmetry. 



To form the equation of the oval of single symmetry of any degree, it is not necessary 

 to go through the process of forming a surface equation and then transforming to new 

 axes. I have only done this to illustrate the theorem that every plane curve is a section 

 of a homogeneous surface of the same degree. 



From the mode of formation of the preceding expressions, it is easily seen that a 

 symmetrical binomial function of x and z, with the highest power of y added, becomes an 

 oval of single symmetry when a definite value is given to z, as in the following illustra- 

 tion : — 



F(x + zf = P{« 8 + 8x 7 z + 28x 6 z 2 + 56xW + 70.x% 4 + 56a:V + 28x 2 z 6 + 8xz" + z s ] ± y s = 1 , 



By making %= 1, we obtain 



F(x + zf = P{x 8 + 8x 7 + 28a 6 + 56a 5 + 70a 4 + 56a 3 + 28a 2 + 8a } + P - 1 = =F y* ■ 



If all the signs of the second equation be changed, then the positive sign of y gives the 

 closed oval, and the negative sign of y the asymptotic form. 



In this equation for any possible value of x, the positive and negative roots of y are 

 equal ; but for a given value of y the roots of x are unequal. 



On these considerations the following methods have been devised for obtaining the 

 curves of single symmetry of any degree (1). In any diametral homogeneous equation 

 in v x v 2 we may take v x = z+ Jx; v 2 = y ; .'. x= (v x — z) 2 ; whence values of x and y are found 

 from v x v 2 for any required value of z. Or we may take v\ = y 2 ; v \ = z 2 + x, whence x = v\ — z 2 ; 

 and the equation consists of even powers of y and uneven powers of x. 



The curve of PI. VI. fig. 7, which is of the form of a rifle-ball, was obtained from (a, 6) 

 by substituting "5 + >Jx for v x after transforming the origin to the extremity of the axis 

 of X. It is of the 12th degree. I might have taken z + x m = 0, or as 3 = v x — z. 



Each of the homogeneous curves, a, /3, e, and £, may be made to furnish by deriva- 



