But in the above case, rince ƒ (x) = 2ax — x2 + a vă - a According to the rule just given, substitute x + h for x, then ƒ (x + 1) = 2ax+2ah - x3. 2xh h' + a √ z3 + 2xh + h3 — a3 which, upon the supposition that == a becomes Expanding √2a + h by the binomial, and representing the co-efficients by A, B, C. √2a + h = (2a + h)a = A + Bh + Ch' + D¿3 + . . . Substituting ƒ (a + h) = a' — ha + aλh3 + aBhi+aChi+... a series which gives the true developement of ƒ (a + 4) but which does not proceed by integral powers of h. CHAPTER VII. APPLICATIONS OF THE DIFFERENTIAL CALCULUS. ON THE THEORY OF VANISHING FRACTIONS. P WHEN a fraction both of whose terms are functions of r becomes when i particular value is assigned to the variable as x = a, it shows that (x — C) is s common factor both of numerator and denominator, and in order to find the real value of the fraction, we must make this factor disappear from one or bust terms. Now, when Taylor's theorem can be applied to erpand ƒ (a + (a+b), we have Hence the rule to find the value of a vanishing fraction. Differentiate both terms of the fraction the sume number of times, until one or other ceases to become 0, on the supposition that a = a. Then substitute a for z in both terms of the fraction, and the result will be the value required. It is necessary to take the second differential co-efficient, because the common actor of the two terms of the original fraction is (≈ — c)'. erefore the true value of the fraction is 0, the factor of the numerator is If Taylor's theorem fails to give the expansions of ƒ (a + h), © (a + h), which will happen wherever ƒ (x), or Ø (x) contains a radical which vanishes for x = 4, we must obtain the expansion by some other method. Substitute. (a + h) for x in both terms of the fraction, and developing by binomial theorem, we shall have Dividing both the numerator and denominator by the lowest power of h Now make h=0 It is useless to take the differential co-efficients of the terms in this case, because they become infinite. Making ≈ = a + h, we find for h = 0 We may here employ Taylor's theorem to determine those terms of the series for which it holds good, we shall thus obtain upon substituting c + A for z dividing by and then making ǹ = 0 we find 1 for the true value of the fraction. When a gives a product ƒ (x) × © (x) of the form 0 x α, Then = Thus when z=1 the supposition that x = a, but by the above process we shall have CHAPTER VIII. ON THE THEORY OF MAXIMA AND MINIMA. WHEN the variable upon which any proposed function depends passes succes. sively through all degrees of magnitude, the different values of the function may form first an increasing and then a decreasing series, or vice versa, and may go on increasing or decreasing repeatedly, and vice versa. That value at which an increase of the function ends, and a decrease begins, is called a maximum, and that at which a diminution ends, and an increase begins, is called a minimum. The essential characteristic of a maximum consists in its being greater than each of the values of the function which immediately precede and follow it; and that of a minimum in being less than both these values. Let y be any function of x in which this variable has attained a value which constitutes it either a maximum or a minimum. Then if x be increased and diminished by an indefinitely small quantity h, the developements of (x + h) and (x − h) will exhibit the values of x, immediately adjacent on each side to that value which renders y a maximum or minimum. Hence it follows from our definition, that the values of y corresponding to (x + h) and (xh) will in the one case be both less than the maximum, and in the other both greater than the minimum. Now, in order that y may be a maximum or minimum, the values of y' and y, which immediately precede and follow it must be both less or both greater than y. .. When y is a maximum or minimum, (y' — y) and (y, — y) must both have the same sign. But when his assumed infinitely small, the whole of the expansions (1) and (2) will have the same signs as their first terms. Hence, (yiy) and (y' - y) cannot have the same sign, unless p vanishes, .. in order that y may be a maximum or a minimum, the condition requisite is that, |