1072 HON. LORD M'LAKEN ON SYSTEMS OF 



which is different from any of the previously given forms. But it has been found, 

 as the result of the computation of values of r and 0, that this curve is identical with 

 No. 10 of the table given below, which is of the form (/3) ; the explanation being that 

 the curve consists of four real and four conjugate branches, and has accordingly four pairs 

 of conjugate diameters. 



As the result of a study of the fundamental forms here given, I have found that there 

 are certain other critical values of P which produce characteristic forms of the equiaxial 

 curves. These I proceed to enumerate. From the drawings and relative tabular places 

 (computed by the method of homogeneous variation) a very complete conception may be 

 obtained of the possible variations of this family of sextic curves and their projections. 



(146). Limiting Forms of the Equiaxial Curves. 



If n, the index number of a curve, be divisible into factors, p and q, a symmetrical 

 function of the p th or q th degree may be a limiting form of the symmetrical curve of the 

 n th degree ; for we have only to raise the equation of the p th degree to the power q, or 

 to raise the equation of the q th degree to the power p to obtain such a limiting form. 



Thus, (l) by raising the equation x 2 =by' 2 = A 2 to the 3rd power, we obtain the circle 

 and the equilateral hyperbola in the sextic form, 



x 6 ± 3* V + Sx 2 y* ± y e = A 6 ; 



where, the upper sign being taken, we see that the circle is a limiting form of the 

 equiaxial curve (a), when P = 3. The lower sign being taken, the curve is the equi- 

 lateral hyperbola, which is thus showm to be a limiting form of the equiaxial curve 

 (e) when P = 3. Similarly by writing x/a for x, and yfb for y, it may be shown that any 

 ellipse or hyperbola is a limiting form of the sextic curve which is the projection of (a), 

 or (e) to the principal axes a and b. 



(2) It might be expected that a symmetrical cubic would also be a limiting form of 

 an equiaxial sextic. This, however, is not universally true. I shall, however, write 

 down the limiting forms obtained by squaring the symmetrical cubics x 3 ± y s = A 3 , and 

 x 2 y ± xy 2 = A 3 . 



These forms, with the equivalent forms obtained by transforming to secondary axes, 

 are as follows : — 



(x 3 ±y s ) 2 ) J« 6 + 6a;V+9«V 



i=A«- j^+^v+^V) 2A6 



j ' {if + GyW + difx*) 



a: 8 ± 2x 3 y 3 + y 6 



(x 2 y±y 2 x) 2 \ lx«-2xY+ *Vl = 2A 6 



x 4 y 2 ±2x s y 3 +x 2 y*) \y< i =2y i x 2 + yW) 



The forms in the second column are limiting cubic forms for a sextic curve referred 

 to a transverse diameter X, and an asymptote Y, or the converse, as is made evident by 

 dividing the equations by x 2 or y' 2 respectively. 



