This devil's curve is also known as the devil on two sticks.
Gabriel Cramer (1704  1752) was the first to investigate the
curve, in 1750 ^{1)}. Cramer was a Swiss mathematician, he is best known for his work
on determinants.
Lacroix ^{2)}
studied the curve in 1810.
And there was a publication about the curve in the Nouvelles Annales de Mathématiques in 1858 ^{3)}.
Many authors place an extra constant before the y^{2 }term, but this is
only a linear distortion of the curve.
It seems to me
that the devil in the name of the curve is from the diabolo game, being the form
of the curve. The confusion is the result of the Italian word diabolo meaning
'devil'.
This makes also clear the meaning of the sticks in 'the devil on two sticks',
these are the sticks used to handle the diabolo.
For a=25/24 the curve is called the electric
motor curve.
The middle of the curve shows the coils of a wire, which rotate by means of the
forces exerted by the magnets around.
The double devil's curve results when combining the curves for a and 1/a.
notes
1) Introduction a l'analyse des lignes courbes algébriques, p. 19 (Genova, 1750).
Cramer mentionned as equation: y^{4}  96 y^{2}  x^{4} + 100 x^{2} = 0, so that a = 100 / 96^{2}.
2) Traité du calcul différentiel et du calcul intégral volume I, 334336 (Paris, 1797; 1810 ed.).
Lacroix introduced a parameter a, in equation: y^{4}  96 a^{2} y^{2} + 100 a^{2} x^{2}  x^{4} = 0, so that a = a(Lacroix) * 100 / 96^{2}.
3) Page 317
4) La courbe du diable, the American mathematical monthly, vol. 33, p.273
