Summation of Series. 405 
fluxion **f can be deduced the fluents of all the other fluxions 
xp, xD, &c.; and in general, if «— is divifible by,n, then 
from the fluent of the fluxion +f can be deduced the fluent of 
the fluxion +%6. 
z. Suppofe f = the terms of the binomial theorem ex- 
panded according to the dimentfions of x, wx. (4+ dx") - = 
att - as "bxt + &ce. beginning from the firft or any other 
terms; then, if a, @, &c. divided by ~ give whole affirmative. 
numbers, will all the fluxions +f, x°6, pf, &c. be integrable ; 
and if the differences of the quantities «, @, y, 5, &c. are divi- 
fible by x, from the fluent of the fluxion «xf can be deduced 
the fluents of the fluxions +*f, wf, &c. 
If p denotes the fum of the alternate or terms whofe diftance 
from each other are m, of the binomial theorem, the fame: 
may be applied.. 
3. If p=a+bx+cx"*; and a, B, y, 4, &c. divided by # 
give whole affirmative numbers, then from fp can be de=. 
duced all the remainder i xp, fi xp, &cc.: and in general from: 
two can be deduced all the remainder. 
To find when the fum of any feries of this kind can be 
found, add together each of the fluents, which can be found 
from each other, and not otherwife, and fuppofe their fum =o ; 
and fo of any other fimilar fluent, and from the refulting 
equations can be difcovered when the feries can be integrated. 
13. If the general term of a feries contains in it more: va=. 
riable quantities, z, v, w, &c.; then find the fum of the feries,. 
firft,, from.the hypothefis that one of them (%)-is only varia-. 
bles. 
