1060 HON. LOED M'LAREN ON SYSTEMS OF 



9. Classification and Forms of Curves considered as Sections of Surfaces whose 



Equations are Homogeneous. 



In this chapter I do not enter on the question of the singularities of curves, a theme 

 which has already been the subject of much learned investigation. My purpose is (1) 

 to discover the different elementary and symmetrical forms of the curves of a given 

 degree, which may be considered as the sections of a homogeneous surface parallel to its 

 principal planes ; and (2) from these elementary forms to show how by variation of the 

 unknown quantities corresponding types of unsymmetrical curves of the n th or given 

 degree may be obtained, and the surfaces traced in series of contour-lines. 



I ought here to point out that the motive of this investigation is somewhat different 

 from that of Mr Frost's valuable work on Curve Tracing. 



In a treatise on Curve-Tracing in general, the exact determination of the locus of the 

 curve is of course unattainable, and only approximate methods are used. 



In the present paper, only those curves are considered whose loci can be exactly 

 traced, by solving these equations rigorously for successive positions of x and y. In the 

 diagrams, which are photographic reductions of the original tracings on diagram-paper, 

 the error at any point ought not to exceed 7 ^ of an inch. 



It is perhaps unnecessary to prove that every homogeneous equation of three 

 variables represents a surface symmetrical about three principal axes of symmetry, which 

 it is convenient to consider as placed perpendicularly to one another. This follows from 

 the consideration that when the homogeneous equation is transformed to polar coor- 

 dinates, it contains only the highest power of r, which in the case of a curve of even 

 degree has always equal positive and negative roots. In the case of curves of uneven 

 degree, the same results are obtained by considering the sign of the arbitrary term 

 indeterminate, — as it evidently ought to be, because by so treating it, we obtain from 

 the equations of uneven degrees forms which are strictly analogous to those of the 

 nearest even degrees. 



This being premised, if in the surface represented by the given homogeneous equa- 

 tion we take for the axis of Z, the direction of the radius vector of maximum length ; 

 and if the surface be referred to this axis Z and to a central plane perpendicular to Z in 

 which angles are denoted by 6, then for every value of 6 and <f> there are four equal 

 values of r corresponding to the four permutations of the positive and negative values of 

 d and </> and also other four formed from n - 6 and <f>. Hence, for any plane through the 

 axis of Z, there are four equal values of r, and the curve is symmetrical about Z and the 

 diameter in the plane XY. By transforming to an axis X coinciding with the maximum 

 radius vector of the central plane and a plane perpendicular to it, similar conclusions 

 are obtained for all diametral sections through X, and also for all diametral sections 

 through Y, the line of intersection of the first and second reference planes. Thus the 

 symmetry of the surface with reference to three principal planes and their intersections 

 is established. 



