SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 



1087 



about the axis perpendicular to Y. To obtain a curve of single symmetry from any 

 central surface referred to conjugate axes XYZ, we have only to transform to new axes 

 x and z, leaving Y unaltered. If we then make z = unity, or any arbitrary value, within 

 proper limits, we obtain an equation in x and y, which is the required equation. If the 

 given central surface have all its terms positive, then the form of the curves of the oblique 

 sections (parallel to an axis) resembles that of the Cartesian oval; that is, it is a 

 symmetrical closed curve without inflexions, but more pointed towards the positive 

 direction of x than towards the negative. 



I shall give an illustration of such a curve of the 6th degree. Let the surface 

 equation be 



X<5 + Y6 + Z 6 = l. 



This, when transformed to secondary axes in the plane XZ has the equation 



In the equation, as first given, let Y remain unchanged, and let the equation be 

 tranformed to axes x and z, each having the inclination 60° to the original plane XY. 

 The formula of transformation is 



X = (x — z) . cos 60° ; Z = (x+z). sin 60°. 



If in the transformed equation z be taken equal to unity, and the equation cleared of 

 fractions, the resulting expression for the plane curve is 



16y 6 = 9 - { 7a; 6 + 39a; 5 + 105a; 4 + 130a; 3 + 105a; 2 + 39a:}, 



where the new arbitrary term, 9, is the difference between z 6 or unity and the arbitrary 

 term of the transformed surface equation. The new plane xy is then inclined at 60° 

 toXY. 



The following values of x and y have been computed : — 



- x = -1 -■! 



2 -3 



-•4 



-•5 --6 



-•7 



-•8 



-•9 



±y =(9/16) 1 / 6 -953 -972 -978 



•976 



•967 -950 



•919 



•881 



•801 



= 9086 















- x = --99 -1 



+ # 



1 



•2 











± y = -561 0- 



±y 



•761 



impossible 











The approximate value of +x when y = is "154. 



The value — x = — 1, when y = 0, is exact. 



If the equation of the derivative surface contains the terms Yx^y 1 + Pa; 2 ;?/ 4 , the equa- 

 tion of the section may contain additional terms of the form y 2 (x i + x 3 + x 2 + x) and 

 y*(x 2 + x), where the coefficients are omitted. 



