2l8 
Mr. Hellins’s Improvement of 
To proceed with the greater clearness, I will here set down 
the fluxion in a general form, and its fluent, in the two series 
which are used in the following particular instance, and may 
be applied with advantage in similar cases. 
3. The fluent of * is — r - - 1 - 
^ + 
+ 
^ + 2 " x m + 3n 
m -f 2 n ■ m + 3 n 
&C. which series, being of the simplest form which the fluent 
seems to admit, was first discovered, and probably is the most 
generally useful. But it has also been found, that the fluent of 
the same fluxion may be expressed in series of other forms, 
which, though less simple than that above written, yet have 
their particular advantages. Amongst those other forms of se- 
ries which the fluent admits of, that which suits my present 
purpose is —=== - — v + 
m »M _L H. TM 4- 2 M. 
n. 2,1. in. X m + 3n _j_ which, to say nothing 
m. m + n. m + 2 n. m + 3 n. 1 — *"1 
of other methods, may easily be investigated by the rule given 
in page 64, of the third edition of Emerson’s Fluxions; or its 
equality with the former series may be proved by algebra. 
4 . On account of the sign - before x\ in the last series, it 
may be proper to remark, that its convergency by a geometri- 
cal progression, will not cease till becomes = 1, or x 
becomes = ; and that, when x is a small quantity, and n 
a large number, this series will converge almost as swiftly as 
the former. For instance, if x be= </-§-, and n = 8, which are 
the values in the following case, the former series will con- 
verge by the quantity x n == */?| s = tt> an ^ t ^ 1 ^ s ser * es by 
quantity = where the difference in con- 
vergency will be but little, and the divisions by 80 easier than 
those by 81. 
