FLU
FLU
rences that are always varying, B cannot be faid to increafe or decreafe at any one conftant rate ; and it is not lb obvious how the Fluxion of A being fuppofed equal to its increment a, the variable Fluxion of B is to be determined. It cannot be fuppofed that the Fluxions and differences are always in the fame proportion in this cafe ; but it is evident, that if B in- creafe by differences that are always greater than the equal
fucceflive differences by which — X A increafes, it cannot
increafe at a lefs rate than — X A ; and it cannot increafe at a n
greater rate than - x A, while its fucceflive differences are al- n
ways lefs than thofe of 1 x A. The Fluxion of A, being ftill
n reprefented by a, the Fluxion of B therefore cannot be lefs than
— X a in the former cafe, or greater then — X a in the latter. n b n
The following propofitions are confequences of this, and will enable us to determine at what rate B increafes, when its rela- tion to A is known.
The fucceflive values of the root A being reprefented by A — a, A, A-f-tf &c. which increafe by any conftant difference a, let the correfponding values of any quantity produced from A, by any algebraic operation (or that has a dependance upon it fo as to vary with it} be B — b, B, B + b, &c. Then if the fucceffive differences b, b &c. of the latter quantity always increafe, how fmall foever a may be, then B cannot be faid to increafe at fo great a rate as a quantity that increafes uniform- ly by equal fucceffive differences greater than b, or at fo fmall a rate'as any quantity that increafes uniformly by equal fuccef- five differences lefs than b. In like manner, if the relation of the quantities is fuch, that the fucceflive differences b, b &c. continually decreafe ; then B cannot be faid to increafe at the fame rate as a quantity that increafes uniformly by equal fuc- ceffive differences greater than b, or lefs than b. Therefore the Fluxion of A being fuppofed equal to the incre- ment a, the Fluxion of B cannot be greater than b or lefs than b, when the fucceflive differences b, b &c. continully increafe; and cannot be greater than b or lefs than b, when thefe fuc- ceffive differences always decreafe.
In the fame manner, if the latter quantity decreafe while the former increafes, and its fucceffive values be B -+- i, B, B — b, &c. then if the decrements b, b &c. continually increafe, B cannot be faid to decreafe at fo great a rate as a quantity that decreafes uniformly by equal fucceflive differences greater than b, or at fo fmall a rate as a quantity that decreafes uniformly by equal fucceffive differences lefs than b. Therefore in this cafe the Fluxion of A being fuppofed equal to a, the Fluxion of B cannot be greater than b or lefs than b. And in the fame manner if the fucceffive decrements b, b &c. always de- creafe, the Fluxion of B, cannot be greater than b, or lefs than b. Vid. Maclauriu's Flux. B. 2. c. I. T. 2. p. 579, feq. As the Fluxions of quantities are any meafures of the refpecf ive rates, according to which they increafe or decreafe; fo it is of no importance how great or fmall thofe meafures are, if they be in the juft proportion or relation to each other. Therefore if the Fluxions of A and B may be fuppofed equal to a and b refpeclively, they may be likewife fuppofed equal to | « and
, , .ma , m b
$b, or to — and
n n
The Fluxion of the root A being fuppofed equal to a, the Fluxion of the fqiiare A A will be equal to 2 A X a.
To demonftrate this, let the fucceflive values of the root be A — u, A, A + u, and the correfponding values of the fquare will be AA-2A« + «k,AA,AA + 2A»-|-m, which increafe by the differences 2.Au — uu, iAu +uu, &c. and becaufe thofe differences increafe, it follows from what has been faid, that if the Fluxion of A be reprefented by u, the Fluxion of A A cannot be reprefented by a quantity that is greater than zAs+m, or lefs than 2Au—uu. This be- ing premifed, fuppofc, as in the propolition, that the Fluxion of A is equal to a ; and if the Fluxion of A A be not equal to 2 A «, let it firft be greater than 2 A a in any ratio, as that of 2 A +■ o to 2 A, and confequently equal to 2 A a + o a. Suppofe now that u is any increment of A lefs than o ; and becaufe a is to u as 2 A a + o a is to 2 A u + o u, it follows, that if the Fluxion of A fhould be reprefented by u, the Fluxion of A A would be reprefented by zAu+ou, which is greater than 2Au + u:i. But it has been fhewn, that if the Fluxion of A be reprefented by u, the Fluxion of A A cannot be repre- fented by a quantity greater than T.Au + uu. And thefe being contradiaory, it follows that the Fluxion of A being equal to a, the Fluxion of A A cannot be greater than 2A1. If the Fluxion of A A can be lefs than 2 A a, when the Fluxion of A is fuppofed equal to a, let it be lefs in any ratio of 2 A — to 2 A, and therefore equal to 2 A a — a. Then becaufe a is to u as 2 A a — a is to 2 A u — u, which is lefs than 2 A« — uu, (u being fuppofed lefs than 0, as be- fore) it follows, that if the Fluxion of A was reprefented by a, the Fluxion of A A would be reprefented by a quantity lefs than 2 A » — u u, againft what has been fliewn. Therefore
(be Fluxion of A being fuppofed equal to et, the Fluxion of A A muft be equal to 2 A a.
The Fluxions of A and B being fuppofed equal to a and b re fpefl.ve ly, the Fluxion of A + B will be a + b, the Fluxhn of A + B\ or of A A + 2AB + BB, will be 2 x A~+~Exa~+b' or 2 A a + 2 B b+ z B a + 2 A b, by the laft article. The Fluxion ofAA-t-BB is 2A1+2BJ, by the fame ; confequently the Fluxion of 2 A B is 2 B a + 2 A b ; and the Fluxion of A B is B a + A b. Hence if P be equal to A B, and the Fluxion of P be p, then p will be equal to B a + A b, and dividing by P, or AB, wefind£ = £ + * . IfQ_A ^ ? ^
a«WofQ,thenQ.B=:A,^-f.|=|, or *pf-J;
and confequently q = 9-"_ 2> — - — A/or" 6- A * 1H A B ~B \Tii B"B~ '•
When any of the quantities decreafe, its Fluxion is to be confi- dered as a negative.
The rule for finding the Fluxion of a power is ufually deduced from the binomial theorem of Sir Ifaac Newton. But as this theorem, though eafily found by induftion, is not fo eafy to demonftrate ; it is proper to proceed upon a principle, the truth of which may be fhewn from the firft algebraic elements, with more facility and perfpicuity.
This principle is, that if n be any integer number, and the fum of the terms E»— ', E— »F, E"— ; F», E»— <F= &e continued till their number be equal to n, be multiplied by E-F theproduawillbeE»_F». For the terms bein' formed by fubducfing continually unity from the index of E and adding it to the index of F, the laft term will be F»— '! The produa of the fum of the terms multiplied by E will be
" ,+ 77* F t ?*~' Fi + E F— ; ' their fum
multiplied by— F gives — E"— ' F— E»— ' F' EF»— '
— F»; and the fum of thefe two produas is E"-L^ F» Sup- pofing E to be greater than F, E" - F" will be lefs than
»E"— ' x h, — F, but greater than n F"— > X t, F For
each of the terms £" -■, E—> F, E— » F>, &e. is greater than the lubfequent term in the fame ratio that E is greater than F, and E'-' is the greateft term ; confequently the number of terms bei ng equ al to n, ,,£'—' h greater than their fum ; and kE-"xE— F is greater than their fum multiplied by E— F, or (by the laft paragraph) greater than E"_F". Becaufe the
laft term F"— ' is lefs than any preceding term, n F»— ' X E F
is lefs than the fum of the terms multiplied bv £._ f or lef* thanE" — F". ' ' els
When n is any integer pofitive number, the root A being fup- pofed to increafe by any equal fucceffive difference- the fuc ceffive differences of the power A" will continually mcreaie For l£A-tf, A. A -I a be any fucceffive values of the root! and A —a , A", A+a " will be the correfponding values of the power. ButA-fV — A" is greater than n A"~ 'a; as appears by fubftituting in the laft paragraph A + a for E, A for F, and a for K — F. In li ke man ner n A' — 'a is greater than A" — n —a". Therefore A+a" — A" is greater than A" — A — a", and the fucceflive differences of the power con- tinually increafe. Tl,c Fluxion of the root A being fuppofed equal to a, the Fluxion of the power A" will be n a A"— 1 .
For if the Fluxion of A" can be greater than n a A"—', let the excefs be equal to any quantity r ; fuppofe equal to the
excefs of •y/A"— ' + — above A, and confequently A+o"~'
= A«— ' -i Then n a x A + o"~' will be equal to
n a A"— 1 + r, the Fluxion of A", Let u be any increment of A lefs than ; and becaufe a is to u as n a x A + /~' t0
» u X A + , it follows (from what has been faid) that if the Fluxion of A be now reprefented by the increment u, the Fluxion of A" will be reprefented by n u x A + o'~\ which is greater than nux. A + u~' , and this laft is itfclf greater than A + k — A". But when the fucceffive values of the root are A — u, A, A + u, thofe of the power are A^T^,", A", A + u , the fucceffive differences of which continually in- creafe ; confequently if the Fluxion of A be reprefented by u the Fluxion of A" cannot be reprefented by a quantity gieater
than A + u — A", or lefs than A'— A — u. And thefe be- ing contradiaory, it follows that when the Fluxion of A is fup- pofed equal to a, the Fluxion of A" cannot he greater than n a A'—\ If it can be lefs than n a A'—', let it be equal to
naA"—r, or (by fuppofing 0— A — ,/a7—' — r - J
n ay
to s« A — o . Then 11 being fuppofed lefs than 0, if
the
