222 
Mr. Woqdhouse on the Integration 
In analytical investigation, after , the tran- 
scendental expression, next, in point of simplicity, is 
dx 1/ ;* hi a particular application, this differential re- 
presents the arc of an ellipse,^ a figure, next, in point of sim- 
plicity, to the circle. 
Many differential expressions depending, for their integration, 
on the integral of dx (— ■ ~ ~:~r ) » it became necessary to exhibit 
it, for all values of x and e. A problem in consequence arose, of 
no small difficulty, named, analogously to the naming of 
f'd.x 
j v7~’ t ^ ie rectification of the ellipse. In the prosecution of 
the researches to which this problem led, it was discovered that 
the hyperbola might be rectified by means of the ellipse, or, to 
speak correctly, and without the employment of figurative 
language, it was discovered that the transcendental expression 
dx J ( Afch ( * v 1 ) might be made to depend, for its inte- ' 
gration, on that of dx J ( ■■ ) er 1. 
The integration of dx J | ) does not depend more on 
the length of an ellipse, than it does, on the time of the vibration 
of a pendulum in a circular arc, or on the attraction of a sphe- 
roid ; but, in each of these problems, it occurs as an analytical 
phrase, an expression in symbolical language, the exact meaning 
of which it is necessary to know. If the meaning be determined 
for one case, it is for all three ; and hence, with the rectification of 
* — — — is as simple an expression : they are considered together in 
V(I— **) 
the following pages. 
•f- The ellipse admits of an easy mechanical description ; and, considered as a section 
of the cone, was admitted by the ancient geometricians into plane geometry. 
