of certain differential Expressions , &c. 237 
This theorem is known by the name of Fagnani’s theorem.* 
When x = v, or the quantity e‘ x J ( j is at its maximum, 
s/=/( 1 ) + 1 — b, 
or/ - (/(/ — /) = 1 — b > 
orE — { E ( t ) — E'} = 1 — b; 
or the elliptic quadrant is divided in such a manner, that the dif- 
ference of the two arcs = difference of the semiaxes. 
From the preceding analysis it is clear, that the computation of 
the integral of dx J (—7—7) is perfectly independent of the ex- 
istence of the ellipse and its properties. But it also appears, that 
the property of the bisection of the ellipse, established geometri- 
cally, ought not to be regarded as a merely curious and beautiful 
property, since, by its aid, the length of the elliptic quadrant may 
be computed. Several other properties, considered hitherto in the 
light of curious and speculative truths, translated, would appear 
analytical artifices, and in computation practically useful. 
By the preceding series, the integral of dx J ma y 
computed, when e is nearly = o or 1. It is necessary, however, 
to possess a method of computing the integral when e is of mean 
value; and the methods I am about to exhibit, are such as to 
supersede the use of the two series ascending by the powers of e 
and b ; in other words, from two similar methods, in all values of 
e between o and 1, the integral may be commodiously computed. 
The principle of the method is this, if df —dx J (— q— r) ; then, 
* This theorem of Fagnani has lately been very neatly demonstrated, by a most 
skilful mathematician, Mr. Brinkley, in the Irish Transactions, by a geometrical pro- 
cess, but not without the use of prime and ultimate ratios. Indeed, the nature of the 
subject is such, that the theorem cannot be established, without the use of the fluxtonary 
calculus, or of some calculus equivalent to it. 
