of certain differential Expressions. 251 
It has been shown, (page 339), that ( F ) ma y 
be transformed into a form such as 
/ 1 f* dv! (A! -\-W dv! /T7v\ 
m . «'+ nJ-jjr + uj (F) ; 
and, similarly, F' into a form as w' u ,, -\-n'J^ r + F". 
Consequently, since J-jjr = ’f~W‘ , we can exterminate ? 
and obtain a resulting equation, such as 
yGF-f- y F -f $F"-\-pu-{- qu"= o ; which expresses the relation 
between the integrals of three expressions similar to 
If jr=i, then w", u'", &c. = o ; consequently, jGF(i)-j“2yF'(i) 
+4^ . F"(i )=o, since, x passing from o to 1 , passes from o to 
its maximum ( 1 ), and from 1 to o ; consequently, between the 
values of x, o, and i, JdF'—zF 1 ^), F'(i) representing what the 
integral F' becomes when u'=i; similarly, Jd F", when x=i, 
=4F'(i). 
Since similar equations must be true for F', F", F"', for F", F 
F IV , &c. as for F, F', F", it is plain that, by a simple process of 
elimination, we may arrive at an equation of the form / 3 F -j-^Fr— x ) 
+t'F( s )-f-7r« , -f £«"-f &c.=o, 1 G, 7T, 1/, £, &c. being constant quantities, 
F("- x ), F("), the two last terms of the series F', F", F" # , &c. 
It is clear also, that we can obtain an equation as / 3 F -f yF'+ 
<£F"-fi=F /// 4-&;c. .... ... =o. 
If, in particular applications, ^ * ^ represents the arc 
or area of a curve, the foregoing results, differently expressed, will 
announce properties subsisting between the arcs and areas of 
similar curves; for instance, when A— i, B == e 2 , the integral 
/- 
dx 
V(x — X*) (X— e x X 1 
expresses the arc of an ellipse, abscissa x, semi- 
