1076 HON. LORD M'LAREN ON SYSTEMS OF 



to an asymptote X ; the polar equations in c and £ are referred to the transverse 



axis. 



Class. 



No. 

 11. 



I. 



Equations of the Curves. 



e 



45° 



f x 6 — x 4 y 2 +x 2 y i — y 6 =1 

 { x 5 y + 6x 3 y 3 + xy 5 =1 



c 



12. 



15° 



(afi-15afy* + 15a?y*-y* =1 



-J =F 6x 5 y ± 20x 3 y 3 =F Qxy 5 = 1 

 ( r 6 .cos(60) = 1 : r 6 . sin(60) = 1 



e 

 or 



t 



13. 



45° 



f £B 6 — 2/ 6 =l =1 



j Sx 5 y + I0x 3 y 3 + 3xy & = 2 



f 



14. 



45° 



( x^+xty 2 — x 2 y* — y 6 =1 

 -< 2x 5 y + 4<x 3 y 3 + 2xy s = 1 

 I r*.cos(20) = 1 : r 6 . sin(20) = 1 



iar 



15. 



22° 30' 



( i\ x h y — 4>xy 5 = 1 

 J4^y_ 4x y 5 =1 



( r 6 . sin(40) = 1 : r° cos (40) =1 



ft 



€ 



16. 



... 



( 4>x*y 2 — 4a% 4 =1 

 | 2x 5 2/ — 4cc 3 7/ 3 + 2an/ 6 = 1 



The mode of variation of these curves is very remarkable ; and it is the more deserving 

 of attention, because it results from the rule of signs (p. 1065) thatyb?" any even degree 



* The identity of the polar equations in the annexed tables with the Cartesian equations is proved as follows : — 

 Cos(20) = cos 2 — sin 2 0; 

 Cos (40) = cos*6 - 6 cos 2 . sin 2 0"+sin 4 0; 

 Cos (60) = cos a 6 - 15 cos 4 0. sin 2 0+15 cos 2 0". sin 1 6 - sin°0. 

 . -. (1) A 6 = r". cos (26) = ^(r 2 . cos ¥e) = (x 2 +y 2 ) 2 . r 2 (cos 2 6 - ain 2 6) 

 = x«+x*y 2 - x 2 y* - y 6 . [No. 14 of Table.] 



(2) A 6 = r°. cos (46) =r 2 . (r 4 . cos 46) = (x 2 +y 2 ) . r 4 . (cos 4 - 6 cos 2 0. sin 2 0+sin 4 0) 



=x° - 5x*y 2 - SxY+y 6 . [No 10 of Table.] 



(3) A 6 = r 8 . cos (60)=r e (coa e - 15 cos 4 0. sin 2 <)+15 cos 2 0. sin 46 - sin 6 0) 



= x*~ lbxy+WxY-y . [No. 12 of Table.] 



Again, observing that — 



Sin (20) = 2 sin 6. cos 6; 

 Sin(40) = 4cos 3 0. sin 6 — 4 cos 6. sm 3 6; and 

 Sin (66) = 6cos h 6. sin 0-20 cos 3 sin 3 0+6 cos 6. sin b 6. 

 .-. (1) A« = r«.sin(2l)=r 4 (r 2 sin 20) = (x 2 +y 2 )2xy 

 = 2x & y4-4x 3 y 3 +2xy\ [No. 14 of Table.] 



(2) A 6 =r«. sin (40)=r 2 (r*. sin 46) = (x 2 +y 2 )(4x 3 y -4xy 3 ) 



= 4x*y - 4xf. [No. 15 of Table.] 



(3) A« = r«. sin (60) = 6x*y - 20x 3 y 3 +6xy\ [No. 12 of Table.] 



