FUNCTIONS. 
thing to be considered is the law of relation which con- 
nects these quantities with each other. “ oe 
this, Lagrange supposes 2 to its value, and k 
come « + Oban any indeterminate quantity which 
is independent of 7. It is evident te hav function 
‘x 4) will then become f(x +7 + 0), and it ap; 
Je te ehs dame remult will be bad if in f (a+ 2) we 
ut i - o instead of i. Therefore also the result must 
be the same, whether we put +o instead of 7, or «+0 
in place of x in the ent 
af fepippPq+eHr + &e, 
By the substitution of i 4.0 instead of in the series, it 
becomes 
fae G+ 0) p+ EH 0) 7 + (é+0)%r + &e. 
which, by expanding the powers of ¢ + 0, and writin 
for the pa of brevity, cay the two first terms of eac 
power, because the arison of these terms is suffi- 
cient for the object in view, is transformed to _ 
(A) 
Siptipt fg + Br tits + &e. 
fop+ 2iog+3Por+4ios+ &e. 
In order to affect the substitution of «+4-o instead of x 
in the same series, we must consider, that seeing the 
function fx becomes f «-ip+2q-ir+ &c. when 
a is changed into x + i, it will become f'2 +0 po? ¢ 
4 or, &e. when « is changed into 24-0. In like 
«manner, if pip’ &e g+iqd + &e rir + 
&e. are what the functions p; g; 7, &c. become when 
wv 4 7 is substituted in them in place of x, and they are 
developed according to the powers of i, we shall have 
by changing z into 0, oa +}. 
pop’ + &. q+ o0q'+ &e. rtor + &e. 
for the developements of the same functions, when 
. 2 +o is substituted.in them instead of x. Therefore, 
by this substitution, the series fe-+ip+iq+ &e. 
will become, by omitting the terms which contain the 
second and higher powers of o; 
The (B) 
4 fabip-Pq+ Br ts + &e. 
; +op+iop! +20q'+Bor + &e. 
This result ought to be identical with the other, inde- 
pendently of the values of 7 and 0, which may be any 
quantities whatever. Now, by the theory of indeter- 
minate quantities, this can only be true when the co- 
efficients of like powers, and products of i ‘and 0, are 
identical ; hence, by comparing the developements (A) 
and (B), we get these identical equations, 
: 2q=p',8r=q', 4s=21', &e. 
from which again we find 
pees GS AP, THF, 8507, &e. 
Remarking now that p is deduced from the original 
function fx, by first substituting x4 7 for x, then de- 
veloping the result f(a +i) into a series, p i 
a to the powers of 2, and lastly, taking for the 
value of p that function which is the coefficient of the 
simple power of i; its origin, and the series of 
tions by which it has been found, may be indicated by 
an appropriate symbol,» We have already put p’, q’, 7’, 
&c. to denote quantities deduced from the functions 
Pr % 1s &e. i pis deduced from «; we may 
similarly denote quantity p by /’ x, that is, by the 
symbol for the function from which it has been deri- 
31 
ved, with the addition of an accent over the character- Functions. — 
istic letter. As the function p or f’ « is derived from 
the function fz, so from the function /” x, a new func- 
tion may be, in like manner, derived, which may be 
indicated by f"«; from this last again another func- 
tion, which may be represented by /” x, may be found, 
and so on: So that, in fact, the functions /’ x, f” x, 
J’ x, &¢. are the co-efficients of i, in the first terms of 
the developements of the functions f(x +- 1), /” (# + 7), 
St" (@ +2), &e. 
e have therefore p = f’ x, and as p’ is the function 
derived from p, as p was from fz, we have p’ =f" x, 
and therefore g = $.f’x. Again, q’ being derived from 
q exactly as p’ was from p, or p from fx, we have 
- 1 
uy —_— i fll” 
g=3f' : and consequently i= zet a, and so 
on. 
Therefore, substituting these expressions in the se- 
ries 
fetip+e®g+ir t+ &. 
which is the developement of f(# + 2), we find 
‘3 
fet daferifet Spee Hose 
ae 
sg. OF hot ath 
This beautiful analytical theorem was in substance 
originaly discovered by Dr Brook Taylor (Methodus 
Incrementorum.)- ange first demonstrated it inde- 
pendently of the fluxional or differential calculus, and 
made it the foundation of his theory of functions. The 
form under which he has ‘given it shews clearly how 
the terms of the series depend on each other, and, in 
particular, how the functions which are the coefficients 
of i may be derived one from another, when the man~ 
ner of forming the first,f’ « from the original function 
f2is known. ~ 
Lagrange calls the function f « the primitive function, 
in feapect ot the functions f’ re: Wa, ke. Thee, again, 
in respect of the primitive function, he calls derivative 
functions (fonctions derivées.) The function f’ x is call- 
ed the first derivative function, or derivative function of 
the first order, or simply the prime function; the func- 
tion, f” «, derived from it, is called'the second derivative 
function, or derivative function of the second order, or 
simply the second function; and again, f”’ x, derived 
from the preceding, is the third derivative function, or 
derivative function of the third order, or third function, 
and soon.” i 
Any function whatever, in respect to that from which 
it is derived, is its derivative function, and this last is 
the primitive function of the other. 
Sometimes, instead of using the characteristic letter 
J, a function of « may be denoted by a single letter y ; 
then, ¥ being used instead of the symbol fz, the sym- 
bols 7’, y"; y, &c. may represent the characters f’ x, 
B i af x, &c. According to this notation, y bein 
any function of x, when x becomes « +h, then y wi 
become 
2 
ytiy + Sy + yyy" + be. 
Since every derivative function. of the first order is 
merely the co-efficient of i in the developement of the 
primitive function fz, when # +47 is substituted in- 
stead of x, the determination of the derivative function 
of any power whatever 2” is in fact the same thing as 
