Œuvres de Charles Hermite publiées sous les auspices de l'Académie des sciences, par Émile Picard.

ON APPLICATION OF TIlE THEORY OF UNICURSAL CURVES. 123
p being rational; making tangx =, we obtain
dtt
t = '(t)(I - t );
therefore the equation
F(v, tu)= o,
must give an unicursal curve; and a solution of this form presents
itself when the integral
X f dt
reduces itself to
x = arc tangt.
Again, lastly, assuming
It. (siax dsinamr)? denoting a rational function of tile sine-amplitude, and its
derived function (this being t.he hypothetis of MM. Briot and
du
Bouquet); it is clear that, writing sinamx t, the derivative cas well as it must be a rational function of t and of the radical
v/(- t) (i - k2i1).
Consequently, the equation
F((v, it) = o
denotes a curve of the species (deficiency) i.
Thus the example XI of these authors,
va -- ( tt2 - ) v; - a 2 ( Ut — ) 4 = o
denoting d ( where at - on writing
V =- ( -2_ — I)t,
gives
Itl2 _ -- -
tV -- ta t 2 ~ t t
t5 -- t9 _ - - a__ _ _
5/ 5 t a-'