of certain differential Expressions, &c. 223 
the ellipse, a problem by itself unimportant, the solutions of other 
problems, are intimately connected; and, with this object in view, 
the determination of the length of a curve line, mathematicians 
have enriched analysis with several curious artifices, and va- 
luable methods. 
To determine the integrals of Jb dx ~yr]> it 1S 
necessary to expand them into series. The difficulty is, to expand 
them into series that converge : the determination of the integral 
of ought to precede that of dx J ; indeed, in 
most of the series that represent the latter is involved 
as a term, and is supposed to be known. The determination 
of each integral presents a curious circumstance, in the cor- 
respondence of certain geometrical properties and analytical 
artifices ; for instance, the theorem for the tangent of the sum 
of two circular arcs, affords, analytically, a means of computing 
the length of the arc ; and, conversely, the analytical artifice* by 
which the integral of is computed; translated, leads to the 
f dx 
* The method of deducing the value of J ^ — — between the values of x, o and 
I, independently of any reference to a circle, is as follows. 
dx dn' du" P dx — r ^ u ' f ^ u " 
V i—x" ~ v'U* + Vi— k" 2, * l 1 C, 1 JV(i-^ ~J V[i—u' 2 ) + jv / (i-«" 2 ) 
-f-C, and, expressing the integrals by their exponential expressions, we may deduce 
(see Phil. Trans. 1802) v! + m'V(i— h' z ) ~ x - Let x — 1 and u' = «" 
Let 
1 . - 2 du' dx 
= 7T co,,se ’ uen,ly 7r^j = vTT^j' 
or twice the integral of 
dx 
du' 
between the values of u', o and — equals the integral of , 
v 
between the 
values of x, o and 1 . Again, put 
du 
dv 
dv' 
. Vl/^l—v' 1 ) + V^—u'i put ll'\ 
= , v — — — and v'z 
.*. as before, 
consequently 
f-rr — ~ 2 f dli — (contained between the values of v', o and — 1 =) 
JVi '-* ) J s/( l - v ) Viol 
Gg 
MDCCCIV. 
