q6o Mr. Woodhouse on the Integration 
rectified by means of an ellipse ; which property is to be reckoned 
curious, I conceive, because the ellipse and hyperbola are sections 
of the same solid cone ; for, otherwise, I do not perceive why it 
is more curious, that an hyperbola should be rectified by means of 
an ellipse, than that any other curve, whose arc = F, (F an in- 
tegral dependent on fdx J (-LziflfL) J should be rectified by ] 
of an ellipse. 
In order to integrate dx J 
means 
7-) by mains offdy J ('—££,)> 
-A'A 
put X 
x — oo , and dx 
then, when z = o x = i, and when z = i 
J (^l-S 
— m*) dz 
— , { putting m = 
[i-z 2 )i V(i-m 2 z 2 ) j* r & 
-k} ; consequently, Jdx J ( — ) (e > i } between the values of 
x — o and x= oo — integral of (l ~ w 2J l . 
& (i -z 2 )i x/[i-m 2 z 2 ) 
values of z = o and z—i. 
now, d { (^$) }=&y 
(i — m 2 ) dz 
between the 
yjx—x 1 ) [i—m z z 2 
(i— z 2 )x (i —m 2 z 2 )z 
'[i—m 2 ] dz 
Hence, — 
(i —z 2 )t ^[i—m 2 z 2 ) 
dz 
( 1_ m ") ,/v'i!-^) (I— 
but, if we put Jdz j = F , fd'z J 'F, 
»=^ 5 S-. 
Then, by equation (a) page 240, 
r dz 
0/(1—**) (*- 
-7W 1 X 2 ) 
F . zF_ 
— 772 * I — 77 ! 
consequently 
( 1 — 777 1 ) rfz 
■sz 2 )z V(i —m 2 z l ) 
