236 
Mr. Woodhouse on the Integration 
or 
or 
1 + 6 2 z z )i (i + z*)- 
df bd z f 
dz dz . db 
i + b z z 1 
(i+z 2 
b z d 3 f 
dz . db z ’ 
b z d'f 
(1 + b* z z )l(i+z z )i 
Now, the differential of 
. (T-i)dz 
' dz . db z ' 
__±__ =dx —L iz££L 
V(i+z z ) (1 + b z z z ) (1 + Z z )i (1 + b z z z )l 
dj , 2 dz 
b% (i+z 5 
b % — 1 
[i + b z z z )i 
dz 
( J. \ 4. dL 4. — 
l V(i + z‘)ti + 6*z a ) n b z ^ b\~i 
+z 4 Vi+b 7 
f 2/ | zdf 
* [i + z z )i[i + b z Z z )i V(i+? 
b.df b z z 
db 
-J -L — l± 
1 z. 2 - />* 
26 <?/ b z d z f 
l^ZTi b z -i db db z 
f 
I- 
/ u . a j o z - 
db ~ b z — 1 V(i+2 2 ') (1 + b z z z ) ‘ - _ 
•••/- ^T-'Tb + i 1 + b ‘) ^ — since ’ when x= 1, 
z= oo,and vi ; +>|i; - j| - , - = ^- = o. 
In order to compute the integral Jdx (— ~^r) (/), When e 
is nearly = 1, by a series ascending by the powers of v/(i — e 2 ), it 
has been found necessary to establish this formula, 
ftej ( 4^1 +/* J -/(’) + £ V 0 ^T- 
Now, this formula, an analytical artifice useful for computation, 
applied to a particular curve, and translated into geometrical lan- 
guage, exhibits a curious property of the curve ; thus, in an ellipse 
whose semiaxes are 1 , ,/ ( 1 — e* ),fdx J ( ) Jdv ^/ ( ) > 
represent arcs (E, E') corresponding to abscissas, x, v; and /( 1) 
is the elliptic quadrant (E (1)) ; hence 
E + E'=E(i 
e - {E( 0 - E'}= exj (^5); 
or the difference of two arcs, one reckoned from the extremity of 
the conjugate, the other from the extremity of the transverse, is 
equal to a right line, represented by e 1 x J ( ’ 1 ^,rp ). 
