Mr, Hellins on the Rectification 
Sect. II. The Methods of computing the Values of the constant' 
Quantities by ‘which the ascending Series differ from the de- 
scending ones. 
1 2. Now the methods of obtaining these constant quantities 
are such as are shewn in my Mathematical Essays , (published in 
1788,) pages 100, 101, 102, &c. to 112; viz. either by com- 
puting the value of both an ascending and a descending series, 
taking fory some small definite quantity, or by comparing the 
values of those series together when y is taken immensely great : 
the former of which methods is more general, but the latter, 
when it can be applied, commonly affords the easiest compu- 
tation. In this section, I shall make use of both these methods, 
as the one or the other is best suited to the case in hand. I 
begin with the use of the latter method, in comparing together 
all the different expressions of the value of z, which are reduced 
to few terms in the case when y becomes immensely great. 
Now, when y is taken immensely great, the value of z in 
Theorem III. Art. 4, becomes barely = ey — d. For, in this 
case, the H. L. , d(^y y ±y )_- L- becomes the logarithm of the 
e y 
ratio of equality, which is = o. And then A is barely = 
s/^eyy + 1) + o = ey + &e. all which terms, 
after the first, vanish in this case; and therefore eeA, which 
occurs in the value of B, becomes barely e 3 y. Moreover, the 
radical expression - ~ {ee l y+ . l) \ which enters into the value of 
2 y y . 
B, becomes barely 
— e3 y . 
; and thence we have B 
— e 3 y + e 3 y 
2 ’ 2 
= 0. And, since each of the expressions gee. 
evidently becomes = o, in this case, and since B has been 
