A Treatise on the Differential Calculus

involves only integral powers of h, a function which in its development requires fractional powers. Another and perhaps better illustration is the following: Let us consider the case of (x2-a2), which is to be developed, when a = a + h, so that the original function = 0 when a = a, and the firstderived is ∞; but as the function may be written in the form (x − a)* (x+a)*, this becomes, when x = a + h, h3 (2a + h)*, the second factor of which may be developed in the ordinary way, and thereby the whole development will consist of terms

of the powers of h whose exponents are of the form n + 2' n being a whole number.

Similarly, if the original function has factors of the form (x-a), that and all its derived-functions are infinite when x = a. Thus suppose we have to develope by Taylor's Series, when x = a + h,

sin x
F(x) =
x2

a2,

this and all its derived-functions become infinite when x = a; but the function may be written in the form

the first factor of which, when x = a + h, becomes and the

second fulfils the conditions of Taylor's Series and may be expanded in the ordinary way; but the resulting development will have at least one term involving a negative power of h, which indicates the cause of the failure, viz. that the function does not admit of development in ascending integral and positive powers of h, which alone is given by Taylor's Theorem.

Secondly, as to the failure of Maclaurin's Theorem. There are two cases corresponding to those above discussed; one when the original function involves factors of the forms - and

+, the former of which is always infinite when x = 0, that is, at the lower value of x, and the latter of which, when differentiated more than m times, will have negative indices, and thereby will be infinite when x = 0. And another case is F(x) = log x, which = ∞ when x = O, cessive derived-functions.

and so do all its suc

Also, if the function to be expanded by Maclaurin's Theorem

as e and all its derived-functions vanish when x = 0, the development of F(x) will be that of f(x) alone, and therefore

will be inexact for f(x) + ex2; the reason of the inexactness

being that ea does not admit of development into a series of ascending integral and positive powers of x, which alone Maclaurin's Theorem gives.

There may also be many other functions which are not capable of development in the form of Taylor's Series, and therefore the student must be on his guard against attempts at development in a form not suited to the function; and the above remarks must be considered as general hints rather than as a secure and scientific enumeration of cases of applicability and non-applicability; the latter is beyond our knowledge, and therefore with the former, however incomplete, we must be content.

We have treated thus at length of the cases where Taylor's and Maclaurin's Theorems fail, because some foreign and most English writers have attempted to raise the Differential Calculus on them as its basis. As far as they are applicable, reasoning founded on them may be correct; but since they are not universally so, it is objected, and validly objected, that the basis of the Calculus is confined within limits narrower than is necessary. And no criteria are known for determining whether functions can be expanded in their forms or not, before such principles of continuity, as those which we have made fundamental, have been elucidated; on the expansion-principles therefore we are left to grope our way in the dark, being uncertain whether the matter which we are discussing is within their comprehension or not.

SECTION 2.-On Functions of two or more variables.

122.] Let F(x, y) be a function of x, y; our object is to find the value of F(x + h, y + k), in ascending powers of h and k, F(x, y) and all its derived-functions being finite and continuous for all values of the variables between x, y and x + h, yk, h and k being finite increments of x and y.

Let us consider the finite increments of x and y to be ht and kt, so that ultimately they may be reduced to hand k by putting t = 1; our object is to expand F(x + ht, y + kt), which we will consider to be a function of t, so that