SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1069 



of a homogeneous curve is always that of the curve deprived of its arbitrary term. Hence 

 yjx = bja is a solution of an equation in either of the forms 



x n x n-2 y2 3,71-4 yi ^ 2/" _ A 



a" ^ aF 2 ~P ^ a" 3 * fr ~ ¥ ~ 



x n x n-1 x n-i y n-i yn-1 y„ 



If we divide the first of these by y n we obtain the form 



ra— 4 



=F=F = 1 



Treating x[y as a single quantity u, we see that an equation of descending powers of u 

 equated to unity is always soluble, if its coefficients constitute a homogeneous function 

 of a and b, in which case ajb is a real root of the equation. 



(136). To find the Secondary Axes of any Curve of Symmetrical Expression referred to 



principal Axes. 



The equation if homogeneous is of the form, 



x n ± A 2 *"- 2 2/ 2 ± A 4 cc n - V ± . . . =F A/f- 4 =p A 2 x 2 y n _~ 2 +y n = 1 . 



In this symmetrical expression (after transformation to secondary axes equally 

 inclined to the primary) every pair of homologous terms produces an expansion of the 

 form (5) or (6) of p. 1065 above. The coefficients of the corresponding terms in the two 

 expansions are equal, and the sum of the alternate columns is zero. 



In the same way for curves of the form 



A^j-A^lVj. x 2 y n ~ 2 y n _ 



A a „± A 2a „_ 2 ^ ± • . . -t" A 2a2 bn _ 2 ± bn - 1 , 



it may be shown, by taking u = xja, and v — yjb, that, when the curve is transformed by 

 the formula for axes equally inclined to the primary axes, the coefficients of the alternate 

 columns disappear, or are neutralised, when cos n dja n = sm n djb n ; or tan 6 = bja. 



The solutions here given are applicable to symmetrical heterogeneous curves in 

 any of the above forms, as may be verified by expanding separately the several homo- 

 geneous parts u-l u 2 , &c, which are of the above form; because in the proof of the Rule 

 of Signs it is not assumed that the equation is homogeneous, but only that it consists 

 of pairs of homologous terms equated to a constant. 



