When r = 0, the formula (8) gives

ƒ(x) = ƒ(z) + (x − z) ƒ' { z + 0 ̧ (x − z)},

0, denoting a number between 0 and 1, not generally the same as : hence, replacing ƒ by p, and observing that, by (8), (x) = 0, we have

......

(9).

0 = p (x) + (x − z) p' {z + 0, (x − 2)} Again, by differentiating (8) with regard to z, we have, cancelling terms in the result which destroy each other,

Consequently we have, from (9), since p(x) = 0,

2)}.

and therefore we may replace, in the formula (7), the expression for the remainder R,, viz.

87. In the demonstration of Taylor's theorem we have supposed that the function ƒ(x) and its derivatives are all of them finite. If, for a particular value a of x, this should not be the case, the theorem then becomes inapplicable, or is said to fail. Suppose for instance that f(x) involves a term of either of the following forms

(x). (x − a)TM. p(x). (x - ajr+w

.(1),

...(2),

m and n being positive integers, and @ a proper positive fraction.

In the case of the form (1), f(x) itself becomes infinite when X = a, and, in the case of the form (2), all its derived functions after the nth. These two forms comprehend the only cases of

failure which can present themselves in ordinary algebraical functions.

which shews that a failure of the theorem, due to a term in f(x) of the form (1), corresponds to the existence of negative powers of h in the true development of ƒ (a + h).

Putting x = a + h in (2), we see that

$(x). (x − a)n+∞ = $(a + h), hn+w

which shews that a failure, arising from a term of the form (2), corresponds to the existence of fractional powers of h in the true development of ƒ(a + h).

88. When the first derived function of f(x) which becomes infinite for the particular value a of x, is of the (r + 2)th order, we may employ Taylor's development provided that we do not carry it beyond the term involving h', and that we take care to preserve the remainder R, which may be evaluated by the formula of Art. (84), a formula which is always applicable when the (r + 1)th derivative, as we are now supposing, remains finite between the limits.

Lagrange's Theory of Functions.

89. The extreme generality of Taylor's series had for a long time attracted the attention of analysts, when Lagrange conceived the idea of adopting it as the basis of a theory of functions, the object of which was to arrive at the conclusions of the differential calculus without introducing the idea of limits or infinitesimal quantities. We will give a brief sketch of the general system of Lagrange, which has been, until within the last few years, so generally adopted in elementary treatises.

Lagrange assumes, in the first place, that any algebraical function f(x + h) can be developed in a series of the form

ƒ(x) + p ̧(x). ha + $1⁄2(x). h3 + $z(x). h* +....

He then proceeds to shew that, from the algebraical nature of functions, these powers of h can neither be fractional nor negative, so long as x and h remain general in form. He remarks that, if the series were to involve a fractional power of h, this term would have several values, and that accordingly, for a single system of values of x and f(x), the function f(x + h) would have several distinct values; a conclusion which cannot be true for all values of x and f(x), precisely as, to borrow an illustration from algebraic geometry, it is impossible that all the points of a curve should be multiple points. Again, if the series were to involve negative powers of h, he observes that the corresponding terms would become infinite when h = 0, and that consequently f(x) would be always infinite, which is impossible, except for particular values of x.

From the above reasoning, then, he concludes that the development of f(x + h) is expressed in a general form by the equation

....

f(x + h) = f(x) + h p ̧(x) + h2 p2(x) + h3 $z(x) + The coefficient p,(x) of h, Lagrange calls the derived function or derivative of f(x), and represents by the expression f'(x). Having thus defined the method of derivation, he determines without difficulty the law by which the functions (x), p(x),... are derived from ƒ'(x), and thus arrives at the formula of Taylor, h

f(x + h) = f(x) +

· ƒ'(x) +

1

h2 1.2

ƒ"(x) +

h3 1.2.3

ƒ''(x) + • • · ·,

each of the successive functions ƒ'(x), ƒ"(x), ƒ"(x),. . . .being derived from the preceding just as f'(x) is derived from ƒ(x). These functions are termed the first, second, third, &c. derivatives of f(x), the order of derivation being indicated by the dashes.

If then in the elementary functions x", log x, sin x, &c. we substitute + h for x, and then expand, by the ordinary pro

cesses of algebra, (x + h)TM, log (x + h), sin (x + h), &c., in series ascending by positive integral powers of h, the coefficients of the first power of h in these developments will be the first derivatives of xTM, log x, sin x, &c. The second derivatives may in the same way be derived from the first, and so on indefinitely to higher orders of derivation. The derivatives of composite functions, which depend upon those of the elementary functions, may then be determined.

The theory of functions therefore, according to the method of Lagrange, resolves itself into a mere algebraical system, to the exclusion of what he considers to be the extraneous idea of infinitesimals or, in the language of Newton, fluxions. For the complete development of this theory the reader is referred to Lagrange's two systematic Treatises, entitled Théorie des fonctions analytiques and Leçons sur le calcul des fonctions.

Within the last few years the logical value of Lagrange's system has been called into question by all writers of authority both in France and in England. The chief objections to his method may be arranged under four heads.

(1) All inductions drawn from developments in the form of divergent series are devoid of solidity, and frequently, as may be ascertained from actual instances, lead to erroneous results. It would appear therefore that Lagrange's method, as involving the consideration of series without regard to convergency, is not entitled to the reputation, which it originally possessed, of being rigorously logical.

(2) The hypothesis that f(x + h) may be expanded in a series by positive integral powers of h, restricts the application of Lagrange's system to the ordinary functions of algebra, whereas the general theory of limits embraces all continuous functions whatever, and involves a code of doctrine which exists independently of its application to any subordinate science.

(3) In the application of the theory of functions to geometry and natural philosophy, the idea of limits cannot really be avoided, although it may be disguised by the artifices of algebra. The doctrine of limits on the other hand, as explicitly involving in an abstract form essential principles of our conceptions of