DIFFERENTIAL CALCULUS. INTRODUCTION We have already seen that any binomial of the form (a + b)" may be expanded in a series of the form whatever may be the value of n; but although this theorem is very extended in its application, algebraists have invented another method for the developement of quantities more simple in its first principles and more general in its application. This is called the method of indeterminate co-efficients: and we now proceed to explain its nature. In order to convey an idea of this method, let it be required a to develope the expression a+ba in a series ascending by powers of x. a' b'x manifest that such a developement is possible, for a It is a+b'x may be put under the form a (a' + b'x)−1, and by applying the binomial theorem to this expression, we shall obtain a series ascending regularly by powers of x. Let us then assume a a+b'x = A + Bx + Cx2+ Dx3 + Ex1 + Fx3 +........... where A, B, C, D, ... • ............... (1) are quantities involving a, a', b', but independent of r, coefficients whose value we are required to determine and which for that reason are called indeterminate co-efficients, although in strict propriety of language they ought rather to be styled co-efficients to be determined. In order to ascertain the value of these co-efficients, let us multiply the two members of the equation (1) by a + b'x, arranging the result according to powers of x and transposing a, we find 0 = Aa'—a+(Ba'+Ab')x+(Ca'+Bb')x2+(Da'+Cb')x3+(Ea'+Db')x2+....(2) We may here remark, that, if we suppose the values of A, B, C, D, . properly determined, the equation (1) must hold good whatever may be the value of x, and in like manner equation (2) also. Let us suppose then x = 0, this last equation becomes Aa' whence we deduce the value of A, viz. a A being equal A = a when x = 0, must preserve the same value, whatever may be the value assigned to x, for, by hypothesis, A is independent of x, therefore whatever may be the value of x, equation (2) reduces itself to 0 = (Ba' + Ab') x + (Ca + Bb') x2 + (Da' + Cb') x3 + (Ea' + Db') x2+ .. or dividing both sides by x 0 Ba+Ab+ (Ca' + Bb')x+(Da' + Cb') x2+(Ea' + Db') x3 (3) Since this equation must hold good whatever may be the value assigned to, let us suppose x = 0, hence Since B must preserve its value whatever may be the value of x, we may suppress in (3) the first term Ba + Ab', which disappears by the value of B, and dividing both members of the equation by x it becomes 0 = Ca' + Bb' + (Da' + Cb') x + (Ea' + Db') x2 + Let us again make x = 0, we have We at once perceive that each co-efficient is formed by multiplying the one If we reflect upon the reasoning employed in the process we have just executed, we shall at once perceive that the fundamental principle of the method of indeterminate co-efficients consists in this, that, If an equation of the form (A, B, C, D, ... being co-efficients independent of x) hold good whatever be the value of x, then each of the co-efficients must separately be equal to 0. In fact, since the co-efficients are independent of x, if we can determine their value by making particular suppositions with regard to the value of x, these values must still be the same whatever value we may afterwards assign to r. But by making x = 0, we find A = 0 and both members of the equation being divided by x, it becomes 0 = B+ Cx + Dx2 + making x = 0 in this new equation, we find B=0 and the original equation is reduced, after dividing both members by x to 0 = C + Dx + separately, and in this manner, we obtain as many equations as there are co-efficients A, B, C, D, to be determined. · ..... This principle may be enunciated under a different form. hold good whatever be the value of x, the co-efficients of the terms affected by the same power of x, in the two members of the equation, are respectively equal to each other. For if we transpose all the terms in the second member of the equation, the equation will be of the same form as that given above, from whence we may conclude that we shall have several examples of the application of this principle in what follows. To expand a*. a may be put under the form 1 + (a — 1) .. a2 = {1 + (a — 1)}* Expanding by the binomial theorem = 1 + x (a− 1) + * (x — 1) (a− 1)2 + x (x — 1) (x —2, x (x-1) Arranging according to powers of r where 1 2 3 (a-1)+.. (a —1)3+ . . =1+{(a−1)—}(a—1)2+}(a—1)3—¦(a−1 )*+.....}x+Px2+Qx3+ · = 1+ px + qx2 + rx3 + p = (a1) § (a — 1)2 + § (a — 1)3 — † (a − 1)+ + . . . . Now, since = (1 + px + qx2 + rx23 + . . .) × (1 + px + qx2 + ræ3 + ...) Performing the multiplication = 1 + 2p.x + (p2 + 2q) x2 + 2 (pq + 1') x3 + ............................... (2) Equations (1) and (2) are identical, therefore comparing the co-efficients of homologous terms which is the series required, it being remembered that p = ( a − 1) — } ( a − 1)2 + } (a — 1)3 — † (a — 1)* + . . . To find series for the logarithm of any number in terms of the number itself, and the base of the system. Let N be any number whose logarthm is x, in a system whose base is a, then { (N − 1) — § (N − 1 )2 + } (N − 1)3 — † (N − 1)2 + . .} + Pm+Qm2 +.. = x { (a — 1) — § ( a − 1)2 + } (a — 1 )3 — § (a — 1 )a +..}+P'm+Q'm2+.. where P, Q, P, Q, ... are independent of m, and since m is indeterminate, we have by comparing co-efficients of homologous terms, (N − 1) — § (N−1)2 + } (N− 1 )3 —..=x {(a−1)—}(a—1)2+}(a—1)3— .. } (N-1) (N − 1)2 + (N − 1)3 .. x= (a — 1) — 4 (a — 1)2 + } (a − · . the series required. To find the logarithm of a number in a converging series. series which is always convergent, since the denominator of any term is greater than the numerator, and from this series the logarithm of any number may be calculated. It appears from what has preceded, that if a be the base of a system of logarithms, then the constant multiplier p is p = (a1) † (a − 1)2 † † (a — 1)3 . . . The system of logarithms in which this constant multiplier is equal to unity, is called the Hyperbolic System, and is generally employed in analytical investigations. The base of this system is usually designated by the symbol E. Since Let To develope sin. 0 cos. O in a series ascending by powers of * Sin. A+ A′0TM! + A"μTM" + where the co-efficients and exponents of è are unknown. Since the supposition = 0 gives sin. ◊ = 0, cos. = 1, it follows that if we make 0 these series will be reduced, the first to 0 and the second to 1. It is manifest, therefore, that none of the exponents of in the above series can be negative, since any term such as BP would become infinite, on the supposition = 0. Moreover, the series for cos. must always have unity for one of its terms, since the whole series must become = 1 when = 0. It is rigorously demonstrated by the theorem of Archimedes that, for the first quadrant, the sine is always less than the arc, therefore, whatever be the value of we shall have where K involves and its powers, and decreases indefinitely as decreases, but never becomes equal to 0. From these considerations it appears that the series for the sine and cosine will be of the form * Both the reasoning and the calculations employed in the following demonstration are somewhat intricate, and, therefore, those students who are not yet familiar with such operations will do well to omit this article upon firct reading the work, and take for granted the conclusion arrived at. |
