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

I24 OEUVRES DE CHARLES IIERMITE.
cement of the Integral Calculus that these find an application,
M. Hermite proceeds as follows: I have remarked that they give
rise to a method of integration of equations of the form
F~d ttJ=treated of by MM. Briot and Bouquet, in the Journlal de i'Ecole
Polytechz iqtue.
Suppose, in fact, that the question is to determine the integral
when it is an algebraic function of the independent variable.
The question is easely resolved in all the cases where the number
which determines the nature of this function is = o; that is, if it
is possible to take rationally
u -= (t), x = -(t).
In fact, from this hypothesis, it follows that
du _ '(t)
dx ~'(t)
is also rational in t; wherefore it is necessary (although not sufficient) that, assuming
dx
the curve
F(v, ut) = o
should be unicursal. Deriving then, from this relation the rational
expressions
= (t), t = (t),
we obtain
dx = d- = ( t) dt
u t)( )
and thence
x -d '(t) dt.
But this integral can always be obtained rationally, and, in the
case where the logarithms disappear, gives the value of x in the
assumed form.
In the case where tt is of the form

it =- (tangx),