the terms of which consist each of a single power of i mul tiplied by a function of x, that is entirely independent of i, the developement shall contain only the positive integer powers of i, and cannot by any means contain either a ne gative or fractional power of that quantity, provided that the value of x he altogether indeterminate. If, however, par ticular values be given to x, then the proposition will not be universally true. Our limits oblige us to refer to La grange's work for the demonstration (Theorie des Fonc tions), which has in some respects been rendered more complete by Poisson, Correspondence sur L'Ecole P.olytech nigues, No. 3.
It being ascertained that the developement off (x+i) has in general the form r+ in which p, g, r &c. are new functions of x, which derive their origin from the original function fx, the next thing to be considered is the law of relation which connects these quantities with each other. To determine this, La grange supposes x to change its value, and become x+o, o being any indeterminate quantity which is independent of i. It is evident that the function f (x+i) will then be come f (x+i+o), and it appears also that the same result will be had, if in f(x+i) we put i+o instead of i. There fore also the result must be the same, whether we put instead of i, or x o in place of x in the develope ment f &c.
By the substitution of i-Fo instead of i in the series, it be comes f r+ &c.
which, by expanding the powers of i-Fo, and writing, for the sake of brevity, only the two first terms of each power, because the comparison of these terms is sufficient for the object in view, is transformed to f x-f-i it+ q + r + s + Ste. o s+ &c.
In order to effect the substitution of x o instead of x in the same series, we must consider, that seeing the function f x becomes f r+ &c. when x is changed into x+i, it will become r, &c. when x is changed into x-Fo. In like manner, if p-1-ip'-1- Ste. y+ i g'+ Ste. r-f-i7"+ Sze. are what the functions p, g, r, Ste. be come when x+i is substituted in them in place of x, and they are developed according to the powers of i, we shall have by changing i into o, 11-1-o p' Ste. q-l-o g' + Ste. r+o r'+ &e.
for the developements of the same functions, when x+o is substituted in them instead of x. Therefore, by this substi tution, the + &c. will become, by omitting the terms which contain the second and higher powers of o, f 11-1- + r s &e.
+op+ i o g' or / Ste.
This result ought to be identical with the other, indepen dently of the values of i and may be any quantities whatever. Now, by the theory of indeterminate quantities,
this can only be true when the coefficients of like powers, and products of i and o, are identical ; hence, by comparing the developements (A) and (B), we get these identical equations, 2 y=fz', 3 r=g', 4 s=ri, &c.
from which again we find q r=4 g', a=+r', &e.
Remarking now that p is deduced from the original func tion f x, by first substituting x-Fi for x, then developing the result f (x+i) into a series, proceeding according to the powers of i, and lastly, taking for the value of p that function which is the co-efficient of the simple power of i; its origin, and the series of operations by which it has been found, may be indicated by an appropriate symbol. We have already put p', g', 7', Stc. to denote quantities deduced from the functions p, g, r, Ste. exactly as p is deduced from x ; we may similarly denote the quantity p by f' x, that is, by the symbol for the function from which it has been de rived, with the addition of an accent over the characteristic letter. As the function p x is derived from the func tionfx, so from the function f' x, a new function may be, in like manner, derived, which may be indicated by f" x ; from this last again another function, which may be repre sented by f" x, may be found, and so on : So that, in fact, the functions f' x, f" x, f" x, are the coefficients of i, in the first terms of the developements of the functions f(x+i),f(x+0,f(x-1-0, Exe.
We have therefore p=f'x, and as p` is the function de rived from p, as p was from f x, we have pf= f" x, and therefore g=if" x. Again, g' being derived from g ex actly as p' was from p, or p from f x, we have g'=if"' x, and consequently r= 213 and so on.
Therefore, substituting these expressions in the series f x-fti r Ste.
which is the developement off(x+i), we find i2 f (x+i)= f x+i x-1- x+— x f i4 . 3 . 4 fiv x &e.
This beautiful analytical theorem was in substance ori ginally discovered by Dr Brook Taylor (Itlethodus lucre mentorum.) Lagrange first demonstrated it independently of the fluxional or differential calculus, and made it the foundation of his theory of functions. The form tinder which he has given it skews 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 manner of forming the first f' x from the original function f x is known.