i 20 Mr. Hellins on the Rectification 
equations marked (£) and (d) in Art. 7 and 9,) which are 
operations equally easy. 
Thus it appears, that all the labour of computing the eccen- 
tricity, the abscissa, and the length of the elliptic arch denoted 
by 'E, and of applying it to the rectification of the hyperbola, 
is wholly unnecessary; and consequently, that that method is 
circuitous , and more curious than useful .* 
Some further Observations on the Rectification of the Hyperbola : 
among which the great Advantage of descending Series over as- 
cending Series, in many cases , is clearly shown; and several 
Methods are given for computing the constant Quantity by which 
those Series differ from each other. 
13. The new series above given in Art, 11, it is obvious, 
will converge swiftly so long as u is small in comparison of 1; 
(which it will be when x is not much greater than 1 ;) so that 
this series will be very convenient for computing a small arch 
of an hyperbola near the vertex, even when e is nearly = 1 : 
and, when e is small in comparison of 1, any arch, how great 
so ever, may easily be computed by it. But, when e is nearly 
= 1 , and uu is greater than this series will converge but 
slowly; and, for that reason, will not be an eligible form for 
arithmetical calculation. In such cases, however, a swift con- 
vergency will take place in some of the descending series 
(discovered by me, and) inserted in the Philosophical Trans- 
* By comparing the expressions marked y and in Article 9 of this Paper, with the 
short paragraph near the top of page 261 of the Philosophical Transactions for 1834, 
the mathematical reader will quickly perceive that Mr. Woqdhouse has there as- 
s^-£ed too much. 
