462 Mr. Hellins on the Rectification 
descending series given in Theorems III and IV. 
which d is 
viz. A x 
aa 
3 a 
2.2.4 
+ 
3-3 5» 
2.2 
.44.6 
The series to 
. 3 3 5 S-7 a& 
2 . 2 . 4 . 4 . 6 . 6.8 ? 
&c. will indeed diverge when a is greater than 1; yet, as was 
observed in Art. 7, it is of that form which admits of transfor- 
mation into another which will always converge. 
16. For the reasons above given in Articles 12 and 13, each 
of the terms A, B, C, &c. in Theorem VI, Ait. 8, vanishes 
when y becomes immensely great, and z is then barely 
= |\/(“ w + aa ) And , since \/ (nu + aa) is, by the notation 
in Art. 1 and 7, = « fi[eeyy -f ee ), which, in this case, becomes 
barely — ey, we have z — ey — d. Here we see that the series 
in this Theorem, and in Theorems III and IV, are always = 
to each other, and consequently differ from each of the ascending 
series by the same constant quantity d , the value of which was 
discovered in the preceding Article. 
17. When y becomes immensely greaff the value of z in 
Theorem VII, Art. 9, becomes barely — u — d . And, since u is 
universally = v/ ( eeyy -j- 1 ), it will, in this case, be = ey ; and 
we shall have z = ey — d, which is the very expression given 
by all the other descending series in the like case. But, when 
the hyperbola is equilateral, as was supposed in Art. 9, a is 
= 1, and we have d = 1-57079632 x : f — -7— + 
3*3 5-5-7 
&C. 
2. 2. 4. 4. 6. 6. 8 ’ 
Moreover, when y is = o, z is also 
therefore, by this theorem, we have o 
X .3 3-5 
O, 
and u is = 1 ; 
2.3 
2.4.7 
2. 4. 6. xi 
FAZ — &c. or 
2.4.6.8.15 ’ 
_JA — — — 3 'Jo 7 — J &c- 
2.4.6- x.x 2.4.6.8.15 
2.4.7 
