SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1061 



A homogeneous surface may have more than one set of axes of symmetry. Some 

 of these may be conjugate diameters meeting the surface at finite points, and some of 

 them may be asymptotic lines. 



In a central equation the axes of reference are asymptotic lines, if the equation wants 

 the highest powers of the three variables ; because then, dividing the equation by the 

 lowest power of any of the variables, suppose x p , we find for the values, y = 0, x=0, the 

 corresponding value z — A/af = oo . 



When a homogeneous central surface is referred to axes of symmetry, its equation 

 must consist either entirely of terms of even powers of each variable z, x or y, or entirely of 

 terms of uneven powers of the same variable ; because then only will the value of the term 

 be unaltered when x is changed to —x. or y is changed to —y. It is of course only in 

 curves of even degree that x and y are both even or both uneven, and therefore curves 

 of uneven degree have in general only single symmetry unless the sign of the arbitrary 

 term is treated as indeterminate. Accordingly, 



1. If the equation of a homogeneous surface is of uneven degree, and consists of 

 terms of even powers of x and uneven powers of y, the axes of reference are asymptotic 

 lines. 



2. If the homogeneous surface equation, being of uneven degree, consists of terms 

 of uneven powers of the variables, the axes of reference coincide with finite conjugate 

 diameters ; but this condition can only be fulfilled if the equation is of the form — 



where the sign A" is indeterminate. 



3. Again, if the equation of a homogeneous surface be of even degree, and consists 

 entirely of terms of uneven powers of the variables, the axes of reference are asymptotic 

 lines. 



4. If the equation being of even degree consists of terms of even powers of the 

 variables, the axes of reference coincide with finite conjugate diameters, unless the 

 highest powers of the variables are awanting. 



5. If, in any of these cases, the surface is expressed by a symmetrical equation, — 

 that is, if the equation consists of pairs of homologous terms, the signs in all the pairs 

 being either like or unlike, — the three axes are equal ; and the surface is also sym- 

 metrical about six secondary conjugate axes, which bisect the angles between each pair of 

 the first set. Moreover, there are two planes through the axis of Z and a secondary axis 

 lying between the axes X and Y, which are planes of symmetry ; and two for each similar 

 combination ; that is, six planes of symmetry in addition to those originally given. 



6. If the equation be of the form f n (x/a, yjb, z/c) = 1, and be a symmetrical function 

 of these ratios, the surface will of course .be a " 3-dimensional projection," or homo- 

 geneous transformation of the corresponding function of x, y, z. It is evident from 

 known principles that all lines and planes of rectangular symmetry will be projected into 

 lines and planes of oblique symmetry ; and the secondary planes and lines will bisect 



