j^ 2 Dr. Waring on 
firft feries, which afcends according to the dimenfions of **, 
terminates, when mn + h ± ± is a whole negative number; the 
fecond, which defcends according to the dimenfions of 
the fame quantity x”, terminates when h ~- is a whole 
affirmative number. The fluent will terminate both ways 
w hen is a whole affirmative and m a negative num- 
n 
i h + 1 
ber greater than 
5. When m is a whole number or =0, the fluent Jf ' a + bx" x h x 
can always be found in finite terms of x , or in the above- 
mentioned finite terms, together with the log. of x; 
which appears from reducing a-\-bx n 
into fimple terms 
> 4. mar— 1 bx* + m . a m ~ z tfx" + See., and multiplying them 
into x h x, and finding the fluents of the refulting fluxions : but 
the feries found by the preceding method will not always ter- 
minate when tn is a whole number, and the feries findable as 
above mentioned; when properly corre&ed, it may be ren- 
dered findable, or, which means the fame thing, the feries 
may be made to terminate. 
6. When the feries which exprefles the fluent of a fluxion 
terminates, we may begin either from one end of the feries, 
or the other ; for example, in finding the fluent of the fluxion 
»»4-i 
a + bx" x x’-’x, either aflume the feries a + bx n x (ax h+I + . 
,@x b+ " +I + yv* +2 ”+ 1 +&c.) or the feries a+bx " + x (A + Bx” +] 
Cx 2 ” + &c.) ; the former, as is before mentioned, terminates 
when 
