455 
of the conic Sections , 
parison of 1, and has an evident advantage over the last, in that 
it converges by the powers of ee, as well as by those of yy ; so 
that its convergency will not cease, till the quantities B, C, D, 
&c. increase in the ratio of i to ee, that is, when y becomes 
equal to, or less than, — f— . This series, therefore, will be very 
useful for the greatest part of the hyperbola, when it is corrected 
by the constant quantity here denoted by d, the value of which 
is attainable several ways, as will appear in the next section. 
6 . These four theorems, or indeed two of them only, are 
sufficient for the rectification of any portion whatever of any 
conical hyperbola. Yet, as I have discovered several other series 
for that purpose, which are more convenient in particular cases, 
and of which some are useful in computing the constant quan- 
tity above denoted by d, (by which the ascending series differ 
from the descending ones,) it may be proper now to give the 
investigations of them also. 
7. Put 1 -f" eeyy == uu ; then will yy be = and 1 -{-yy 
=• mi + ee — i _ __ the notation in Art. 1, where ee was put 
= aa J r 1,) 
vu ‘ aa •, and therefore y' ( 1 -j- yy) == 
ee 
and thence 
x -f- e e yy 
i+yy 
eu 
\J ( uu-\-aa ) 
Moreover, y will be = 
UU 
e V (uu — x ) 
u uu 
, and we shall have y s/ 
1 + eeyy 
1 + yy 
u u 
u u 
y [uu— I ) X V [uu 4 -aa) 
V(uu— I) X-/(l + 
aa 
uu 
f/ [uu — 1 ) 
- x : 1 — 
3-5« e 
a a * 
2 uu * 2.42*4- z,/\.. 6 u 6 
u u 
+ Now the fluent of 
,, -—is s/(uu — 1 ) ; and, if the fluents of —rr'—-, ——-A 
&c ' are denoted b y A > C, &c. respectively, we ' 
shall have 
3 N 2 
