FUNCTION. 
wh panied 7 ie,that i is, can be taken lefs, taking the quan- 
tity at its leatt. 
Let now ¢ x be reprefented by a* en D"¢ Q a (A x)”, or 
p* A". a 
» @ 6 (Ae) = ce ae . (Ax)* (T), and confe- 
pri 
Aw Q # 
quently, next term (T") is, T’: aS. “Dp .. = 
a ee and all the fucceeding values of after the 
ae ( A) continually decreafe ; — in this cafe it is 
elear that A » can be taken 7. i whatever is, or can 
be = lefs than the leaft value of ” " : 
et @ xbe reprefented ah . By > PL. x (which fun&ions 
belong to the es me’ *, @ = @*-*), then Oy 
Ox. and confequently, 
nt é 
Diviee eo ce ars which quantity is always 
e 
between — and , and confequently, in this cafe A x can 
be taken 7 Sala 2) wl whatever n is, or can be taken lefs 
= the leaft value of oe - Ue. 
"Hence i in dw it is ues re A scan be taken of a mag- 
nitude fuch, that any one term, as D" 6 x. (Ax), fhall ex- 
ceed the fum of all the facceeding al 3; and a fortiori 
fhall exceed that fum, the {mall er 
ence cof. w, are compre- 
hended by the fymbol 9 x, the priponts n will ftill be true, 
fince the analytical exp reffions for thefe funtions are formed 
by means of the ex ponential c. 
The above propofition i is true of - any feries, as A + Bx 
+ Cx + oe in which it-can be fhewn that a is 
Aras 
a 
7 —— r is. Since, then, x. (the araieeaey 
quantity) can be taken £ 
which fea : increafes bases the increafe of r, and becomes 
infinite ; but fuch ao cannot. occur in thofe cafes wherein 
it is oe ed apply the a ana viz. that the arbitrary 
quantity (# or A «) can nfuch, that ah one term _ 
fhall exceed ne fum .'of at . = faceeding tern 
For inftance, if A + Ba +C#e4& ie eas. x 
#16263, 0" + &e.. ten Date Lets a 
+ Boo 
r+; which: becomes infinite ew . does, ind ee ra 
eamnot be taken £ — but then fuch a feries can 
“never = ‘for it sould ‘be abfurd 2 propofe to deduce, 
’ for inftance, the value of. the bina of a curve from a 
- feries whi cannot be made to conv 
- 
e. 
_ curves aS 
Let y be ihe ordinate of:a- curve, and let it be expreffed 
by the funtion of the abfciffla 9 «, then « becoming 
A dv 
mens there are eries in 
DAR ye 
termine, the conditions under which the contagt of © 
aA 4 
w+ Ax, y becomes y+ § a es 7. ( x) 
4+ &c. let x be the its of another curve, an° let 
u = ft, tbeing an abciffa taken in the fame line of the ab- 
ciffas that x isin, then ¢ becoming t+ Af, ubecomes uw + d # 
Att 
da 
] ae (A#)*-+ &c. Suppofe now the curves to 
have a common point, and for eee ig es let each ab- 
{cifla a gaa from the fame 
7 ae 
ou curves is Fach that betweeen them no other curve pee 
en 
feed 
+ yn Aw +e 
=u ~~ «Ax + ——-_— 
dw .2.dx? 
the courfe of the rs of which v is the ordinate, paffes 
between the two other curves whofe ordinates are y and wu, 
the difference y' — v', or Ay — Ad, ought to - lefs than 
dv 
tae + &c: and bse 
y'—u', or A y — Au, however {mall A x is, or 
oD —- 
ay _ dx dw/- 
ke dy es £2). (a me es . 
a oe ahs I 2c ire) Oo +Gs a 
= ty): (A x)) + &c. (for iy. ees 
wtf os “), however fmall A «& is, or cee by A xp 
(t- ) ta.Ae +5. (Sx) + ke. is always 
Ax + b'.(Ax¥ 4 &c. (a, a’, b, b', &c. being put 
for we ahem of the powers of Ax “ys however fmalf 
Aexi is, which is evidently impoffible, except “. =o 
x 
or d y= for the difference of the two expreffions, to 
cit (@— 4) Set G—5) (Ast 
‘wit, 
eer ores 
d x 
thallbe greater than(a — a‘) Ax 7 — 5) (A ay 
&c. may be made pofitiveby taking A » fach, that af 
d x: 
&e. whi ch, b what has precede d y po offible; hence, 
between the courfes of the aes curves in ens the ordinates 
on point c a exc 
mie “ad v) equals ae differential d ae f the re ina 
dy 
Again, fuppale I= ts de ZorD. ys i or : + ts 
da? : a a 
iF or D*. y = -—~ or. D*. uw, then between thefe two. 
urves no other curve (whofe equation is v = nn & 
the 
ton 
étind ough the common point where 
ordinatesare equal, can 
d 
pafs except - ts e . ie 
7 dw <dew d x =" < . 
For if it - | ni then y'— v’, or + Bg es 
(v +A 7 sn = v by ie othelis 7 Ay —aAv 
— sae C ieaee {mall the interval (A a 
bet i ee ie 3 that is, putting for A’y, Au, 
a ‘i fhe. are equal to, to wit, 
D.y 
