FUNCTION. 
Haw whole proéefs beeomes burthened with all the imper- 
feGtions to which that original theory is liable. All this will 
be more diftin@ly feen hereafter ; what is faid at prefent is 
only to introduce the fubjet, by apprifing the reader of the 
objec of the feience. 
The: notation we fhall adopt is the fame as that ufed by 
Mr. Woodhoufe, after M. D’ Arbogalt. It differs materially 
from that of Lagrange. 
- denotes a isa difference, 
, the differe 
5, the differential co-efficient, 
3, ed variation ne 
faa he nase the co-efficient of the 
term: ae i s found by multiplying the earth 
(m) of the fir . term into that » its ind eing 
minifhed oes ; th refore, denotes ae epeaton 3 
e m Signi ma” a nifies — 
r Dia”, is that three fimilar operations 
to be ma a. vis firft on a”, the fecond on the power of a 
tha at refults after the firit operation, the third on the power of 
a, which remains after t the fecond ; hence D* a” ignifies 
mD'?a"—, of m . m—1-.D a", or m.m—1. m—2. 
q" 
Similarly, D D......Da™ (# we and nim) fignifies m 
(m—1) (m—2) sesme(m—2+1 
Or,if DD D........D (2). be dbridgedly reprefented by 
oe D* a” fignifies m(m—r) (x 2— 2) eivaressvseeces (7% —2-+ 1) 
e Dial, genet 3 Dia’, 4.3.2a3 
D2 1", m. eae fa) im 5 
I ener e 
Dp? 174, ~ 3 e 3 i-sor 2 1-4 or 3; 
D Uy. pai wage Ij, Or La or Ze 
3 3. 3 27 27 
Dia" —m.—m—1.'a-™, or m. (m+1) 
—(m + 2). ‘ 
The e propofition affumed by Lagrange, aad which forms 
the bafis of his whole work, i is this, that if @x be any func- 
tion ora sate a variable quantity: A and x changes its 
value ec x + Z the Q (% + #). may be 
for ied oF vefolved into feries ed this form, ¢ x + 
+O?+ Ri’ 'y &e, which the co-efficients of the 
18 the origina 
ould fuch a general propofition as this be fatiefagtonily 
demonftrated, it would, no doubt, lead us at once a great 
The binomial theorem, and other feries 
2. 
Ss 
ut 
fhewn ee this incon ai eneral, 
and that no. fuch theory. of feries can be eftablithed Bors 
‘Shall preclude the neceffity of any farther examination of par- 
ticular forms. Now the 
unétion of x-or x comprehends, 
under - its general ignition, a variety of combinations, 
fuch as x”, a*, log. x, xy Cc € re- 
fore, effentially: at aoe) in an elementary treatife to confide: 
thefe forms, and to. fhew in what manner, hat re. 
ftrictions, they may be included under any. general expreffion. 
Bhe v well-known binomial theorem we fhall take for granted, __ + 
but not asa propefition ra — ans abitract principle, 
pa as en refult of in ding in ee reat =) 
fare on infpeCtion and: itl. (See. Manni 
oa whe ee to give it BOE eee wien 
as capable of being deduced from eee of motion, as 
in many treatifes on fluxions, feem to have been led into the 
theo orem, and that theorem afte ane demon ree, ca 
application of ‘thofe very rules which could not themfelves. 
have been devifed without the previous renee ledge of the 
theorem to be — . 
ut, however this may be, let us at prefent take the 
truth of thistheorem for granted, and proceed to a farther 
oe of the fubject. 
—For the purpofe of a more commodious notation, 
;> — ———- Kc. 
1.2. 
a+ x)” being ee “fumed 7) . + > a™ x + ta a” 
x?+Dix', &c. the next expreffion to be confidered is 
O@x= a*e 
Dn 
= 
reprefents = 
Now, in the expreffion a* there is nothing indicative of 
any operation that can be performed in a manner analogous. 
to thatof @+x", fom peg muft be ies to change 
its =] for = parpote ne o fimple is, te. 
make 1 += * ma - e put under this form, a* = 
(1+ (a — 1)" 2 by the preceding theorem, " where 
@=1, X=a—1, m= x, a* = (1+ (a—1))*= 14x 
(2a—1) Epo a eaans 7) (4 — 1)? 
pegs ole G22 1, &e. &ee 
ca 62.3 
=1ts((a—t)—4 
(a@—1)? +3 (a— ryp—k 
(a—1)' or A x 
. ee a 5 ((4 — 1) — (4— 1) + (a= Eel, Fs 
= 1 A,» 
in which feries the. quantities vw A, A,, A,,» are conftant 
co-efficients, but unknown. To jes ae ihe law of their — 
connexion or dependance on each other, increafe x by the 
oo Ae ee Bs 
then 1+A(x+2%) 4 A(x + 2) aed Get a)"s 
= x epanng the powers of x + 3, ftopping at "the two frit 
eae +A(e +2) +A, (st axet &e.) + A, 
(3 + 3x°2 + &e. 
Aa—t (x” +a x" 7's + &c.) + A (et + (a + 1} 
xs + &a 
a ada = (1+ Axn+A e..A  x™ 4 &e.} 
x (1+ Az+A, “A at &c. } 
ST A(s +2) + A AY eet A oe? 
+A’waztA A. wz. -A An ley 
Compare the correfponding terms. in the two ‘expanfions, 
d 
pot 
Ss 
_* 
2A, = A’, and A, 3A, = AA == and Ag 
| =A AorA An 2A 
1.2. 3 m—tb 22 n—~i r 
Ay 
L.2.3-4 
“A2 2 3d , 
Hence, a® = r+ Ax+ a, Al x =s A} xt 
1.2 1.2.3 1.20364 
If « =1 
—— 
