CHAPTER V.

ON INVERSE FUNCTIONS.

In the preceding trigonometrical expressions, the sines, cosines, &c., have been considered as functions of the arcs; but we shall now treat of the interse functions, and consider the arcs as functions of the sine, cosine, &c., and investigate their differential coefficients.

A peculiar notation has been adopted to distinguish inverse functions. The arc whose sine is x, is represented by the symbol.... sin −z; the arc whose cosine is z

Here the direct function is x = sin y; and, therefore,

cos-1; tanz;

log

(1.)

dr

dy

(2

In the preceding expressions the radius of the arc is unity; but they may be readily adapted to radius 7, by considering that and dare numbers;

therefore the numerator and denominator of each differential coefficient most be of the same dimensions. Hence, to radius r the formulas now investigated are as under.

We may now investigate the differential coefficients of a few of the more complicated inverse functions, as in the following examples:

Ble e

= 2b (a + bx) (a' + Y′x)3 + 36′ (a + bx)o (a' + bz)

&

dy

• We suppose the logarithme in ‘● taken m the system where p = 1.

• This signifies log. log. s, or the logarithm of the logarithm of r. So log. ↑ is log, log, log. x, er *he logarithm of the logarithm of the logarithm of x. Any generally log,s= log. log, log.

◄ terms, z

...

CHAPTER VI.

ON THE GENERAL FORM OF THE DEVELOPEMENT OFƒ (≈ + k).

IF f(x) be any function of x, and if we substitute x+h for z, where his any indeterminate quantity; and if we develope ƒ (x + h) in a series according to powers of h; then, so long as no particular value is assigned to x.

1. The series can contain no negative or fractional powers of h.

2. The series will be of the form

f(x+h) = fx + P.h+Q. h2 + R. h3 +...

where P, Q, R, &c. are functions of x only.

For, if any term, such as Th‣ could enter the expansion, since he is suscep tible of μ values, we must have μ values of (x + h); but while a remains indeterminate, f(x) must contain the same radicals as ƒ (x + h); and .: ƒ (x) must have μ values: substituting.. the values of ƒ (2) successively in the values of f(x + h), we shall have in the whole 3 values of ƒ (x + h). Thus ƒ (x + h) when developed will have μ2 values, and when not developed it can only have the same number of values as ƒ (2), i. e. μ values which is impossible, except for particular values of x.

Next, to ascertain the form of the developement of f (x + h)

If we wish to ascertain what part of this function is independent of h, we have only to make h = 0, which reduces it to ƒ (x), so that ƒ (x + h) = ƒ(1) + a quantity which disappears when h = 0, and which must.. be multiplied by some positive power of h, and since no fractional power of h can enter, this quantity must be of the form Ah, A being a function of h and a, which does not become infinite when h = 0, thus we have

But since A is a new function of ≈ and h, we may in like manner separate

the part which is independent of h.