468 
And thence 
Mr. Hellins on the Rectification 
+ 
e v' (t + 1 ) = 7-5545,396, 
B 
2.4c 3 
3 -S p 
2.4. 6. 8 e 7 
= 0-0010,341, 
= 0-0000,024, 
+ 7-5555,761 
— 0-1205,849 
+ 
+ 
+ 
A 
ze 
3C 
2.4.65 s 
3 5 - 7 E 
2.4.6.8.105® 
= 0-1205,452, 
= 0-0000 ,395 
• = 0 - 0000,002 
— 0-1205,849 
and % = 7-4349,912 — d. 
But, by the foregoing part of this article, % = 0-6360,768 ; 
we therefore have d = 7 4349, 91 2 — 0-6360,768 = 67989,144. 
22. With the value of a above given, viz. 7, we see a swift 
convergency, both in the ascending and in the descending series ; 
but, if a were given = ^3, (which is as small a value of a as 
need be used in these theorems, for this purpose, because if it 
were less than, or even = 4/3, the value of d might be com- 
puted by one series only, as was observed in Art. 15,) each 
of the series would converge but slowly, in this case, being 
= -§- ; to remedy which, as the terms of each of the series have 
the signs -{- and — alternately, it would be expedient to com- 
pute a moderate number (from six to ten, as the case shall 
require,) of the initial terms of each, and then to transform the 
remainders into other series, which should converge by the 
powers of instead of the powers of y. This increase of 
convergency in the geometrical progression, assisted as it would 
be by the decrease of the coefficients of the new series, would 
enable us to get a result accurate enough for all common uses, 
by computing ten (or fewer) terms of each of the new series. 
