12 Proceedings of Royal Society of Edinburgh. [ sess . 
If we turn the axes through 45°, the equation takes the form 
4{x 2 + y 2 )x 2 y 2 - 9 a 2 {x 2 + y 2 ) 2 + 24 a\x 2 + y 2 ) 
- 16a 6 = 0 . . . (4), 
which is more convenient for some purposes. 
Since the curve is symmetrical with respect to the original axes, 
and also with respect to the new axes, which are the octant lines 
of the old axes, it is immediately evident that it must consist of 
eight congruent portions. 
For the intersection with the cc-axis, taking equation (3), we get, 
putting a = l, for brevity, 
x 6 - 9x 4 + 24x 2 -l6 = 0, 
that is, 
(x 2 - l)(x 2 - 4) 2 = 0. 
Hence the intersections with the ^c-axis are 
(-2,0) his, (-1,0), (+1,0), ( + 2,0) is 
Similarly the intersections with the y-axis are 
(0,-2) his, (0,-1), (0, + 1), (0, + 2) bis. 
We may trace one of the eight congruent branches of the sextic 
by causing 0 to vary from 0 to 7 t/ 4 in the formulae (1). If dashes 
denote differentiation with respect to 0 , we have 
x = (2 cos 3 0 sin 20-3 sin 30 cos 20)/ cos 2 20 ; 
y = (2 sin 30 sin 20 + 3 cos 30 cos 20)/ cos 2 20 ; 
y _ 2 sin 30 sin 20 + 3 cos 30 cos 20 # 
x 2 cos 30 sin 20 - 3 sin 30 cos 20 ’ 
x - yx If = 3/(2 sin 30 sin 20 + 3 cos 30 cos 20). 
e 
X 
y 
x' 
y' 
y'/x' 
a? - yx’ly' 
0 
1 
0 
0 
3 
00 
1 
7T 
IT 
0 
2 
-6V3 
4V3 
-2/V3 
3 
7 r 
T 
- oo 
00 
00 
00 
-1 
3/\/2 
