of certain differential Expressions , &c. 255 
or $/=/( 1 ) + 1 °r/— {/( 1 ) -/ } = l —6,] 
or f-m = -LdL.. 
J 2 2 
This, expressed with reference to an ellipse, announces that the 
difference between an arc of an ellipse (abscissa = » ■ "■■■■■ ) and 
V{i +b) 
half the quadrant of an ellipse, equals half the difference of the 
semiaxes. 
Similarly, the difference between/ and may be asssigned, when 
u'z 
V(i+b’) 
J—Jdxj u"—\ y and, consequently, when 
= 1 T 7 - X J (tESi - j ; thus, supposing the value of x to be a, 
when u"= 1 ; since, 
df= . du'+ — .J±£L + 'T 5 Itf- 
J i + e' 1 (i + e') (i-J-e ) L 2 2 1 2.(i + e') J U" 
<r 
(i+e) (i + e") 
{ 
(!_/) (i+e") 
rr „\ _ e' I I « 
3 \ x — a ) — l+e r y(i + 6' } -r (1+0 (i+O 
i-e" i ritf , /" 
*•(■ + «')// U " ' (•+«■) (! + «') ’ 
and f f 1 1 — r (— ') C+O , <■-*■> ) , r^' . 4/-Q 
anay (ij | 2 . 2 + 2 . (1+0 * I y U" + (1+0 (1^0 » 
••• 4/ (■*=«) -/(!) = — ' - 
+ e' ‘ V(i + 6) « (i + e') (i + e") * 
Now, 1+6'= «'= ^jrj, and e"=^; 
consequently, 
4/— /( i )= 2 (i— v/^) 1/(1+^) + (1— v'&r, 
or/-j/(0= (, -^V (,+t) +(-^f. 
Now, to determine x, we have u'= |~lj= . x v / / 1 7" ^--; ) ; 
consequently, putting v / / *[- -~ t: 2 --)=m, 
*•= or, putting for m% e', their values 
LI 
1— ?/ 
MDCCCIV. 
