SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 



1083 



attempt to illustrate all the varieties. The following illustrations of curves of the 6th 

 degree of two homogeneous parts include the most characteristic forms : — 



(1) x 6 +3x i y 2 +3x 2 y i +y<> = x 2 -y 2 } 



r 4 = cos 20 j 



(la) x e +3x i y 2 +3x 2 y i + y 6 = 2xy j 



r* = sin 20 J 



(2) x 6 + 3x*y 2 + 3x 2 y* +y 6 = x i - 6x 2 y 2 + y* \ 



r 2 = cos 40 j 



(2a) x^+SxY+SxY+y^^y-ixy 3 \ 



r 2 = sin 40 j 



(3) x 6 - 6a; V + 6xY -tf = A%x 2 - f) 



The following Tables contain the computed places for the symmetrical half of a 

 foliation or loop of each curve : — 



Equation (1). 



= 



0° 



5° 



10° 



15° 



20° 



25° 



30° 



35° 



40° 



44° -47 



45° 



T = 



10 



•996 



•985 



•965 



•936 



•895 



•841 



•765 



•646 



•363 



o- 



From 45° to 135° values of r are impossible. 



From 135° to 180° we obtain the above series reversed. 



Similar results from 180° to 360°. 



The curve consists of two loops, and there are two inflexions at the centre. 



PL III. fig. (1) is this curve, and fig. (2) is a projection of it. 



Equation (la). 



= 



0° 



5° 



10° 



15° 



20° 



25° 



30° 



35° 



40° 



45° 



r — 



o- 



•646 



•765 



•841 



•895 



•936 



•965 



•985 



•996 



10 



(la) is therefore (1) transformed to secondary axes, which are the tangents at the 

 central point of inflexion. 



Equation (2). 



Equation (2a). 



= 



0° 



r-5 



4° -5 



7°-5 



10°-5 



13°5 



16°5 



19°5 



22°5 



r — 







•323 



•556. 



•707 



•818 



•900 



•956 



•989 



10 



VOL. XXXV. PART IV. (NO. 23). 



8 B 



