1068 HON. LORD M'LAREN ON SYSTEMS OF 



(2) Where the central equation (being of the 4th degree) also contains terms of the 

 2nd degree P(x 2 + y 2 + xy) the terra in xy disappears by transformation, and the trans- 

 formed expression consists entirely of even powers. The new axes are accordingly 

 conjugate axes. 



More generally an equation of any even degree, consisting of the highest terms of x 

 and y, and of one other term of even powers, may, in like manner, be immediately 

 reduced to the projection of an equiaxial curve. Such a projection always has a pair of 

 secondary axes, which are the diagonals of the circumscribing parallelogram. 



13. Diameters in Central Curves of Higher Degrees. 



For an equation of a degree higher than the fourth, secondary conjugate axes cannot 

 in general be found. Because, if we transform and expand as before, there are for every 

 even degree above the fourth more than two terms of uneven powers ; and it is impossible, 

 unless some relation amongst the coefficients be given, to determine 6 so as to make the 

 coefficients of more than two of the uneven terms vanish. The required relation is 

 easily found. The equation must be a symmetrical function of x/a and y/b. This being 

 premised, if 6 be the angle between X and x, the inclination of the secondary axis, x, to the 

 primary is given by the relation, tan 6 = b/a. This will he made clear by two examples. 



(13a). To find the Secondary Axes of the Curve X n ja n ±Y n /b n = 1. 



(a) Taking first the upper sign, and transforming to axes equally inclined to the 

 primary, by the formula X. = (x — y) cos ; Y = (x + y) sin 6, we find for the transformed 

 equation the expression 



^^\ x n - nx'^y + V^rX y x n-^ - &c. I + ?H±A i x n + na»-*y + &c\ = l. 



Let cos n 6/a n = sm n 6/b n , or tan 6 = bja; then all the uneven terms disappear, and the 

 equation is accordingly referred to conjugate diameters, which are equal in length, and 

 symmetrically placed with reference the principal axes. These diameters coincide with 

 the diagonals of the circumscribing parallelogram, whose sides are parallel to the principal 

 axes of the oval of the ?* th degree. 



(b) If the equation be taken with the negative sign, it is the terms of even powers 

 which disappear in the transformation, and the secondary axes found are asymptotes of 

 the hyperbolic curve. 



(c) It is to be observed that the values above found for the inclination of the 

 asymptotes of homogeneous symmetrical functions equated to an arbitrary term, are 

 solutions of the relative functions equated to zero. Because the equation of the asymptote 



