1078 HON. LORD M'LAREN ON SYSTEMS OF 



± 22° 30', and the first real branch is contained within an angle of 45°. The first 

 conjugate branch and the second real branch are contained within angles of 22° 30' ; 

 and the second conjugate branch (bisected by Y) is contained within the angle 45°, 

 and so on. 



When P= 15 there are six real and six alternate conjugate branches all equal, each 

 contained within an angle of 30° (fig. 3). When P exceeds 15, we have a series of pairs 

 of unequal curves (which have not been fully investigated), but are probably the con- 

 jugates of the preceding set. 



Returning to the neutral form of the equilateral hyperbola, and varying P by 

 diminishing it indefinitely : — If P>0 and <3, x and y can only become infinite for 

 = ±45°; and we have a series of equilateral forms consisting of two real and two 

 conjugate branches. The variation of P between these limits only affects the quadrantal 

 eccentricity. The form x 6 —y 6 = 1 is the limit between the forms (e) and (£). In PI. II. 

 fig. (1) the curve which is nearest the centre is No. 13 of the Table ; the curve furthest 

 from the centre is the limiting equilateral hyperbola; and the intermediate curve is No. 11. 



In the series (£), where the signs of the two intermediate terms of the equation do 

 not follow in alternate order, the curves are inflexional and equilateral, the only 

 asymptotes being the secondary axes. 



In Plate II. fig. 4, the curve which is nearest the centre is No. 14 of the Table. 

 The curve next it, having the same pair of asymptotic axes, is traced from an equation 

 of the same form with a different coefficient, (P=15). The curve which has the axis of 

 X for one of its asymptotes is evidently a limiting form of the same series, and is No. 16 

 of the Table. Its minimal radius- vector corresponds to 



= tan" * -5 = 26° 34', nearly. 



The variety (8) resembles (£) in its forms and inflexions, but is not equilateral, as 

 (£) is. One of these forms is figured, PI. IV. fig. 4. Its equation is 



x 6 +x i y 2 +x 2 y i — y 6 = l ; 



and for the curve figured (P = 1) the inclination of asymptotes to axis X (which depends 

 on the value of P) is 53° 37', nearly. 



All the curves here traced have been computed by the tangent formula, which is the 

 best for studying the transitions from one of the enumerated forms to the other or 

 others. 



General Results. — It is evident that the results which have been obtained are in the 

 main independent of the degree of the symmetrical homogeneous equation. For 

 equations of curves of even degree, referred to axes of symmetry, these results may be 

 generalised as follows : — 



(l) If all the terms are positive, the curve is an oval of fourfold symmetry, entirely 

 concave to the centre, and having the circle as a limiting form. 



