Placing the second member of (4) equal to the second member of (3), and dividing both by y — %, we have, + ply — 2)2 1+% M + &c. = M + N(y + 2) (1 + 2)2 (1 + x) + P(y2 + yz + z2) + &c. Since this equation is true for all values of y and 2, make % = y, and we shall have, M 1 + y = M + 2Ny + 3Py2 + 4Qys + &c. (5) which is an identical equation; clearing of fractions, we have, M = M + 2N Y + 3P y + 4Q | y8 + &c. + M + 2N + 3P Making the coefficients of the like powers of y in the two members equal (Art. 178), we have, Substituting these values of N, P, Q, &c., in equation (1) and factoring, we have, Log (1 + y) + &c.,) (6) y: = M + 2 3 4 which is the logarithmic series. The quantity M is a constant that depends on the base of the system. It is called the modulus. In the Napierian system M = 1, and the logarithmic series in this case becomes 191. Any equation containing but one unknown quantity, and whose exponents are whole numbers, may be reduced to the form, grande + pan-1 + 2-2 + &c. + 2? + ta+u = 0 (1). In this equation, n is a positive whole number, but the coefficients p, q, &c., may be either positive or negative, entire or fractional, real or imaginary. The method of reducing equations to the form (1), is analogous to that given for reducing equations of the second degree to the form, 22 + 2px = 9; and since the reduction can always be made, we shall hereafter, in speaking of equations, suppose them reduced to the form of equation (1), unless the contrary is expressly mentioned. Roots. 192. Any value of x, either real or imaginary, which, if substituted for x in equation (1) will satisfy it, that is, make the two members equal, is a root of the equation. It has been shown that every equation of the first degree has one root, and that every equation of the second degree has two roots; we shall assume that every equation of the ner degree has at least one root, either real, or imaginary. Properties and Transformations. 193. In this and the following articles it is proposed to demonstrate the most important properties and transformations of equation (1), article 191. First Property. If a is a root of equation (1), the first member of that equation is divisible by x - a. For, if we divide the first member of equation (1) by a a, and continue the division till a remainder is found that does not contain X, and if we denote that remainder by n and the quotient obtained by m, we have, Now, if a is a root of the proposed equation, it will reduce the first member of (2) to 0, when substituted for x; it will also reduce the first term of the second member to 0; hence, n is also equal to 0, that is, the remainder is 0, and consequentfy the first member is exactly divisible by a – a, which was to be shown. Second Property. 194. If the first member of equation (1) is exactly divisible by x - a, then is a a root of the equation. If we divide the first member of equation (1) by 2 - a, as explained in the last article, the remainder n will be equal to 0, and equation (2) will reduce to the form, 201 + pan-1 + &c. + tx +u = (x - a)m. (3). If, in (3), we make x = a, the second member reduces to 0; conseqnently, the first member also reduces to 0, which satisfies equation (1); hence, a is a root of (1), which was to be shown. It follows from the preceding propositions that we can ascertain whether a polynomial containing x, is exactly divisible by x — a, by substituting a for æ in the polynomial : if the result is 0, the polynomial is divisible by X - a; if not, the polynomial is not divisible by — a. Third Property. 195. Equation (1) has as many roots as there are units in n, and it has no more. It is assumed that the equation has one root; let that root be denoted by a: then will a a be a factor of the first member, and the first term of the other factor will be an-1; the exponents of x in the succeeding terms of the second factor will be less than |