of certain differential Expressions, &c. 225 
smallness of e', e", are readily computed,) translated into the 
language of geometry, expresses a curious relation between the 
arcs of three ellipses, the excentricities of which vary according 
to a certain law. 
Hence it appears, that there are two different methods by 
which the analytic art may be advanced ; either by artifices 
peculiarly its own, or by aid drawn from the properties of figures 
and curve lines; if, for instance, Fagnani's theorem be proved 
for an ellipse, by processes purely geometrical, then, such a 
theorem, expressed in analytical language, becomes immediately 
a means of computing the integral of dx \ / ~~ e _*z ; or if, by 
reasonings strictly geometrical, a relation can be established be- 
tween the arcs of three ellipses, whose excentricities vary ac- 
cording to a certain law, then, by expressing such a relation in 
the signs of algebra, the integral of dx j —73^) may be 
computed by means of the integrals of du' J ( a °d °f 
du" J ; which integrals can be found more readily 
than the original integral, by reason of the quicker convergency 
of the series into which the differential expressions may be 
expanded, e' and e" being less than e. 
One main object of the present paper is, to exhibit the integral 
of dx J f ) for all values of e, and to reduce other 
integrals to it. Much has been already done on this subject. 
The researches of mathematicians on the length and comparison 
of elliptic arcs, are extended over the surface of many memoirs ; 
yet I hope to have something to add in point of invention, and 
more in point of arrangement and simplicity of expression. 
The labours of future students will surely be lessened, if it be 
Gg 2 
