1062 HON. LORD M'LAREN ON SYSTEMS OF 



those parallel to conjugate planes and lines, but will not bisect the angles between the 

 principal diameters and principal planes. 



7. Paragraphs 1, 2, 3, and 4 may be applied to plane curves by suppressing the 

 element Z ; 6 also applies to plane curves, and it will be shown that the inclination of 

 the secondary axis, x, to the principal axis, X, is given by the relation, tan 6 = bja. In 

 the further discussion of the subject I shall use the term " Diametral Equation " to 

 express the equation of a curve when referred to axes of symmetry. If an equation 

 containing only even powers of the variables x and y be also a symmetrical expression, 

 the curve has fourfold symmetry, because the symmetrical form of the equation shows that 

 the axes of reference are equally inclined to a second pair of conjugate axes. There are 

 then eight points at which the value of R is either a maximum or a minimum. This is 

 a property which is not lost by projection. A Symmetrical Diametral Equation is an 

 equation which is itself symmetrical ; where therefore the curve is equiaxial and has 

 fourfold symmetry. 



10. Classification of Central Curves of the Form ¥(x, y) n = A". 



A central function of two variables equated to an arbitary term may be either 

 homogeneous or heterogeneous. In the first case, the equation may represent either 

 a central section of the general homogeneous surface, or a section taken parallel to a 

 principal plane of any homogeneous surface whose equation contains only the highest 

 power of Z. In the second case, the equation represents a section taken parallel to a 

 principal plane of the general homogeneous surface. Reference is made on this point 

 to the preceding part of the paper. 



If we begin by considering homogeneous symmetrical forms, or forms which are the 

 projections of these, it is evident that the equations must be composed by the multipli- 

 cation of factors of the forms, 



(x , y\ v . (& _,_ 2/ 2 Y ■ ( xn + y"Y 



\ab) ' V W ' "* \«" b n ) ' 



The number of possible symmetrical equations is, however, very much less than the 

 possible permutations of such factors ; and it is not difficult to see that the required 

 number is that of the permutations of the positive and negative signs in a symmetrical 

 equation containing only even, or only uneven powers of x or y. From the preceding 

 remarks it is seen that a diametral symmetrical equation represents a curve which has 

 two pairs of conjugate axes, each axis bisecting the angles between the axes of the other 

 pair ; and that such axes are either asymptotes or finite conj ugate diameters. 



(1) Oval Forms. — If the given equation is homogeneous, and if the four axes of 

 symmetry coincide with finite diameters, the equiaxial curve is generally a symmetrical 

 oval entirely concave to the centre. In this species, if the original equation consists of 

 terms of even powers, the equation of the curve when transformed to secondary axes as 



