1088 HON. LORD M'LAREN ON SYSTEMS OF 



If in the original equation y 6 be taken negative, the surface will be a sextic hyper- 

 boloid of one sheet, y being the axis. Taking a section whose inclination to the original 

 plane XY is 60°, we obtain the same expression for the plane curve as that last given, 

 except that the terms on the right side of the sign of equality have their signs changed. 



The following values of x and y have been computed for this hyperbolic curve of 

 single symmetry : the first value only being approximate, those from *2 to 2*0 being 

 exact. 



+ x = 



•154 



•2 



•4 



•6 



•8 



10 



20 oo 



=F3/ = 







•80 



114 



1-35 



1-54 



1-72 



260 oo . 



The values of x of the immediately preceding table give an opposite and dissimilar 

 branch. 



The two curves are shown in figs. (1) and (2) of PI. IV. 



(19a). To find a Symmetrical Expression for the Oval of Single Symmetry. 



Referring to the figures of the curves, since for every pair of equal positive and 

 negative values of y there axe two values of x, it is always possible to inscribe a square 

 in such a figure. Let the oval be referred to the diagonals of the inscribed square as 

 coordinate axes. Then the equation must satisfy three conditions : — ( 1 ) The uncombined 

 terms of x and y are all terms of even powers, otherwise the values of +x and —x would 

 not be equal when y = ; (2) the equation is a symmetrical expression, because the axes 

 of reference are equally inclined to the axis of symmetry ; (3) the terms are all positive, 

 because, according to the rule of signs (above), it is only positive pairs of even terms which, 

 when transformed to bisecting axes, produce exclusively even powers ofy, as must be the 

 case here. There is a fourth condition. I may here anticipate what is proved in the section 

 on radial coordinates, that the algebraic equations of these curves, when referred to their 

 axes of symmetry, contain no uneven powers in the even terms; i.e., these even terms are 

 of the form x 2p y 2q , and we have already shown that the transformation to axes having the 

 inclination 45°, does not introduce uneven powers. Hence (4) our symmetrical equation 

 may consist of pairs of the terms x°, y G ; x i y 2 , x 2 y 4 ; x 2 y 2 and x 2 , y 2 together with composite 

 uneven terms. If an equation satisfying these conditions be made homogeneous by sup- 

 plying powers of z, it is seen that the axes of the plane curve lie in principal planes 

 of the homogeneous closed symmetrical surface, and that the origin of the plane curve is 

 in a diameter of this surface. 



Unless all the uneven terms are present, the oval will be inflexional. See figure (3) 

 of Plate IV. 



Similar results are obtained for the hyperbolic curve of single symmetry. 



These curves are best investigated by means of radial equations from two foci, as 

 given in the sequel. The origin or point to which the last equation is referred has no 



