1074 



HON. LORD M'LAREN ON SYSTEMS OF 



P 1 is the coefficient of the intermediate terms when the equation is transformed to 

 secondary axes, the arbitrary term then being T 6 . 



No. 



P. 



r. 



Equation of Curve. 



1. 



4 



i 



4x y 2 + 4<x 2 y =1 



2. 



31 



Jh 



x 6 + 3 IxY + SlxY + y« = 1 



3. 



15 



(i) 1 ' 6 



x 6 + 15afy 2 + 15x*y* + y e =l 



4. 



7 



(h) m 



x 6 + 7x*y 2 + 7x 2 y 4 + y G = l 



5. 



3 



i 



x 6 + Sx*y 2 + 3x 2 y i + y e =l 



6. 



1 



(2) 1 ' 6 



x 6 + x*y 2 + x 2 y i + y % = \ 



7. 







(4) 1 ' 6 



x 6 + +2/ 6 = l 



8. 



_i 



2 



72 



x 6 - \x*y 2 - \x 2 y i + y* = l 



9. 



-1 



QO 



x 6 - x^y 2 — x 2 y i + y 6 = l 



10. 



-5 



-1 



x 6 — bxty 2 - 5«V+2/ 6=1 



No. 5 is the limiting circle. Nos. 1, 2, 3, 4 are the transformed equations 9, 8, 7, 

 and 6, with the curves turned round through an angle of 45°. On referring to Plate I., 

 where the numerals attached to the curves are those of the first column of the table, it is 

 seen that starting from the circle, as P falls from 3 to 0, the curve approaches more 

 nearly to the circumscribing square (PL I. figs. 6, 7). For all negative values of P 

 between and — 1 the curve is inflexional (fig. 8), the secondary axis becoming more and 

 more elongated, until at the value P= — 1 it passes into the continuous equilateral 

 hyperbolic (fig. 9). 



For values of P from — 1 to — 5 the curve is the discontinuous hyperbolic, where the 

 angle between asymptote and axis ranges from 45° to 22° 30', or 90° to 45° between the 

 asymptotes. 



In the diagram the curves 1, 2, 3, 4 are seen to be 9, 8, 7, 6, diminished and turned 

 round through 45°. 



For the value P= — 5 (No. 10 of the Table) we have four equal discontinuous hyper- 

 bolics, having angle between asymptotes = 45° (PI. II. fig. 2). The intervening angular 

 spaces may be made to contain four equal and similar conjugate curves by changing the 

 signs of all the variable terms in the equation. The equation of No. 10 referred to 

 secondary axes contains only uneven powers (see No. 15). This curve also has the polar 

 equation, r 6 cos 40 = A 6 . 



For negative values of P exceeding 5 we find hyperbolics of greater eccentricity, which 

 are the conjugate curves of the series found for values of P between — 1 and —5. The 

 equation of these may also be obtained in another form, from the last-mentioned series, 

 by changing the signs of all the variable terms. It is easily seen that a similar series of 

 curves are obtained from the form (y), because one negative term suffices to make the 

 oval inflexional. 



The annexed table contains the places of x and y computed by the homogeneous 

 method for curves 6 to 10 of the preceding table, so far as necessary, viz., from 0° to 45°. 



