FUNCTION. 
ehich, under certain conditions, the curves are to be made 
the reprefentat ive 
— line, : = called the line of the hee 
pee pair! taken, be ca 
he abfeilta, 2 by «: the feveral es 
pe eee . the different values of x, let ftraight lines 
‘be continually drawn, making a certain angle with the 
if a thefe ftraight lines are to be called 
ordinates, and the line or figure m which their extremities 
are continually found, is generally to be called a curve 
line 
The values of ne are to be Aieouel by an equa- 
tion between y an hus y == @«, and if » be increated 
by Az, ene en by ce aes methods y + A y becomes 
(2 
dy d’ a. 
Ae I. eae Fe (Ae) 
and fince se “value of y + Avis to ae pene 
ait ferie smuft converge. For grea eater fimplicity in the — 
g demon fir oa the ordimates are fuppofed ta 
eg above definition of a curve line, the 
locus of the hanes of the ordinates mult, in one in- 
ight line; for the function ¢« being of 
the + 4 (a and 4 conftant quantities) the ex- 
ey of “al the cae vill be found in a ftraight 
line. 
Hie re are —— Bhs sear ie it is known, relative to 
t been fatisfaétorily eftablifhed, and 
nieyand accuracy in ise demonftrations 
of fuch propution is Rogtiee ted e definition that 
e line as is een makin 
eing demonftrated. Thus, . 
nm any two ordinates be Ax, then 
ae the next (y') feparated a an Meal 
ean vy y- Ax t Dy (Ax) + Diy (Aa) + 
Again, the value of another pani Manca y and 
% + Ay at an interval dx 
2 
ris y + 
D?. y (ex)? + &e. 
oy. 
But if the line joining the Ae 
“of gy and y! were a ftraight line, theny + D.y. de 4+. 
Dy. (dx)? + &e. ought to = y + sory D. 
- dae + ~ .y dx”. Ax 4+ &c. which it evidently does 
not, except » - J» Ds, - 3, &c. = 0; that is, it does not, 
the cafe of the curve ’ belonging to anequation, as y = 
+d wae excluded. 
r to prepare the way for the demonftrations re- 
‘lative a ‘tangents, radi of Sabir agae lengths of curve lines, 
&c. it is neceffary to eftablifh this ale ae namely, 
that in the feries ¢ # + +D¢ar.4xe+D Do gw (Say 
+ &c.; Aw may be taken of fuch a magnitude, that any 
term D™ gx (A — fhall be greater than the fum of all 
the fucceeding ter . 
ihe met a ey of fluxions and of limits, I have alread 
mentioned that this propofition, or one equivalent to At is 
required ; for to prove that the limiting ratio of 1: a #"~* + 
-(#-1) o- ne te (a—1) (#2) 
2” a 1.2 
: naw", it is neceflary to thew, that by dimi- 
Vv. ‘ 
ax 
x3 . (A x)’ 
ities wes (A xv)*. 
nithing as sae ratio may be brought nearer . the ratio 
He ‘any affignable quantity, - confe- 
ane it is mecelleey to fhew within what limite t the magni- 
a—1 
tude of the reje@taneous quantity = "tA 
a.n—l.a—2 be 
a a"—3 (A x)? + &c. is contained; 
Ir.2. 
¢ « being a funtion of x, when w is increafed to x +- 2 
¢ aa a: cee becomes v+ Dox. Av+ & 
+D .(Qx)" (Aa) + &e. Now 
if there are two ie terms ue Cae (Ax)", ial 
av YNiter ‘Pe ra 
Ga. (A x)*tt fuch, that £ De a is greater than —— 7 ae 
whatever s is, then Aw = be taken fo. fmall, that any 
term, as D™ ga. (A x)”, fhall be greater than the fum of 
» then 
ig 
“DF: px 
Dow. — + D°tt ox. (A — is LD" OH 
(A2)" (r+ 
Again, ue ie “(Ax)" + Det Gae (Ax)"* is Z 
D" ox. “(A x)" (1 + 4), and fo on, every term being 
lefs “2 half the preceding term, and pe aie Dm 
Gx. (Ax)™ 4 + Dt oun. (A wx)” "On. 
(Ag)"? 4 &ais 2D ge. (As™ (1 +h 4+ 54 
=) 
I~ & 
2; and therefore the fum of terms 
after D™ dv. (Ax)™ muft be lefs than D™ Ox. (Ax)™ 
The propofition then will be demonftrated if it can be 
D+! g x 
all fucceeding terms; for take Aw 2 
3 + Kee) 5 confequently, Zz = Pa. (Ax)™ ( 
fhewn ‘that , however great m, is never greater 
than a finite en quantity ; and in order to afcertain 
this point, reference muft be made to the 
Firft, with regard to 
m. (m — 1) im — as 
“e—@—9) 
a“ me 
™m 
ad . Te2eQsecett 
(Ax)", call this term T; then the fi Jing term (T’) is T’. 
ne 
m—n Ax DT! Ow 
—, and confequently Dos i 
a att q * Pr 
which quantity, it is ie dees not “exceed any- sfignable 
a 
Pere sat whatever yalues of n 
quantity, for it << 
yo (always an ines) be taken, the get namely, 
1 
o> mutt always be contained between — “ and boy 
a+4+°1 
Hence it follows, that Aw can be takea < & 40 
{mi — 8 
whatever 
3M 
