1070 



HON. LORD M'LAREN ON SYSTEMS OF 



14. Sextic Curves of the Homogeneous Form F(x, y) 6 =A e . 



The equation is understood to be referred to principal axes when it consists of terms 

 of even powers only ; but in the case of the symmetrical oval it will be seen that this 

 description applies to each of the two pairs of conjugate axes ; and there is, geometrically 

 speaking, no reason why either pair should be considered principal axes preferentially to 

 the other pair. The curves of this class do not pass through the centre. 



In order to obtain fundamental forms, symmetrical equations are first to be 

 considered. Of these there are in strictness only four species, corresponding to 

 the four symmetrical combinations of the positive and negative signs of the terms. 

 There are, for the sixth degree, two other forms, (y) and (8), in which the coefficients 

 are symmetrical, but the signs are not symmetrical. In the form (y) the extreme 

 terms are positive, and the intermediate terms have unlike signs, the order of the signs 

 being + + — +. In the form (8) the extreme terms have unlike signs and the 

 intermediate terms have like signs, the order being + + + — . The forms obtained 

 by changing all the signs of the variables are, of course, the same curves. Also (y) is a 

 variety of (/3), and (8) is a variety of (£). As the equations are symmetrical, the equal 

 coefficients of the highest powers disappear by division ; the equations of the equiaxial 

 curves of the sixth degree may then be written — 



ai<5 + Pay + Pay + y 6 = A G 

 a 6 - p«y - Pay + y G = A c 



x e + p^y - Pay + y e = a 6 

 x 6 + Pay + P^y - y 6 = a 6 



a 6 - Pay + Pay - y 6 = A 6 

 a 6 + Pay - Pay - y 6 = A 6 



(a) 



08) 



(7) 



0) 

 (•) 

 (0 



Referring to p. 1062, where the principle of classification is indicated, (a) is the Sextic 

 Oval ; (/3) is the Inflexional Oval, passing into the Continuous Hyperbolic ; (e) is the 

 Discontinuous Hyperbolic, consisting of alternate real and imaginary branches ; (y), (8), 

 and (£) are Inflexional Hyperbolics. 



(14a). To find the Equations of the Equiaxial Curves referred to Secondary Axes. 



The coordinates of the original equations being denoted by capitals, if, in the 

 formula of transformation, we were to make Y=(x+y) M f%, we should obtain negative 

 values of A 6 in the transformed functions (e) and (Q. Therefore, let X = (x + y) J\ ; 

 Y = {x — y)J\, and transform to bisecting axes. 



In the transformation of equations (a) and (ft), we have for the sum of the 1st and 



4th terms, 



l/4{a 6 + 15ay+15ay + 2/ c }=X 6 +Y 6 ; 



