If, in the series to be summed, the alternate terms (viz. the 2d, 4th, 6th, &c.) should be required to be taken under signs contrary to what they have in the original series given; the reasoning and result will be no-ways different; only, instead of making p³ + q³ + r³ (or pⁿ + qⁿ + rⁿ, &c.) = +3 (or +n), the same quantity must, here, be made = -3 (or -n). From whence, pⁿ being = -1, qⁿ = -1, &c. the values of p, q, r, &c. will, in this case, be the roots of the equation zⁿ + 1 = 0.
It may be proper, now, to put down an example, or two, of the use and application of the general conclusions above derived. First, then, supposing the series, whose sum is given, to be x + x² ⁄ 2 + x³ ⁄ 3 + x⁴ ⁄ 4 ... + xm ⁄ m + xm ⁺ ¹ ⁄ m + 1 + xm ⁺ ² ⁄ m + 2 ... + xm ⁺ ⁿ ⁄ m + n + xm ⁺ ⁿ ⁺ ¹ ⁄ m + n + 1 +, &c. = - H. Log.(1-x) (= S); let it be required, from hence, to find the sum of the series (xm ⁄ m + xm ⁺ ⁿ ⁄ m + n + xm ⁺ ²ⁿ ⁄ m + 2n &c.) arising by taking every nth term thereof, beginning with that whose exponent (m) is any integer less than n. Here, the terms preceding xm ⁄ m being transposed, and the whole equation divided by xm, we shall have 1 ⁄ m + x ⁄ m + 1 + x² ⁄ m + 2 + x³ ⁄ m + 3, &c. = - 1 ⁄ xm × H. Log.(1 - x) - x + ½x², &c. ⁄ xm. In which value, let px, qx, rx, &c. be, successively, substituted for x (according to prescript) neglecting intirely the terms x + ½x² ⁄ xm, as having no effect at all in the result: from whence we get - 1 ⁄ (px)m × Log.(1 - px) - 1 ⁄ (qx)m × Log.(1 - qx) - 1 ⁄ (rx)m × Log.(1 - rx), &c. Which multiplied by xm (the quantity that before divided) gives - 1 ⁄ pm × Log.(1 - px) - 1 ⁄ qm × Log.(1 - qx) - 1 ⁄ rm × Log.(1 - rx), &c. = n times the quantity required to be determined.
But now, to get rid of the imaginary quantities q, r, &c. by means of their known values α + √αα - 1, α - √αα - 1, &c. it will be necessary to observe, that, as the product of any two corresponding ones ((α + √αα - 1) × (α - √αα - 1)) is equal to unity, we may therefore write (α - √αα - 1)m (= rm) instead of its equal 1 ⁄ qm, and (α + √αα - 1)m (= qm) instead of its equal 1 ⁄ rm: by which means the two terms, wherein these two quantities enter, will stand thus; - (α - √αα - 1)ⁿ × Log. (1 - qx) - (α + √αα - 1)m × Log. (1 - rx).
But, if A be assumed to express the co-sine of an arch (Q), m times as great as that (360° ⁄ n) whose co-sine is here denoted by α; then will A - √AA - 1 = [156](α - √αα - 1)m, and A + √AA - 1 = (α + √αα - 1)m: which values being substituted above, we thence get
- A × (log. (1 - qx) + log. (1 - rx))
+ √AA - 1 × (log. (1 - qx) - log. (1 - rx));
whereof the former part (which, exclusive of the factor A, I shall hereafter denote by M) is manifestly equal to - A × log. ((1 - qx) × (1 - rx)) (by the nature of logarithms) = - A × log. 1 - (q + r).x + qrx² = - A × log. (1 - 2αx + xx) (by substituting the values of q and r): which is now intirely free from imaginary quantities. But, in order to exterminate them out of the latter part also, put y = log. (1 - qx) - log. (1 - rx); then will ẏ = - qẋ ⁄ 1 - qx + rẋ ⁄ 1 - rx = - (q - r) × ẋ ⁄ 1 - (q + r) × x + xx = - 2√(αα - 1) × ẋ ⁄ 1 - 2αx + xx = - 2√-1 × √(1 - αα) × ẋ ⁄ 1 - 2αx + xx; where √(1 - αα) × ẋ ⁄ 1 - 2αx + xx expresseth the fluxion of a circular arch (N) whose radius is 1, and sine = √(1 - αα) × ẋ ⁄ 1 - 2αx + xx; consequently y will be = - 2√-1 × N: which, multiplied by √AA - 1, or its equal √-1 × √1 - AA, gives 2√1 - AA × N; and, this value being added to that of the former part (found above), and the whole being divided by n, we thence obtain - AM + 2√(1 - AA) × N ⁄ n, or 1 ⁄ n × (-co-s. Q × M + sin. Q × 2N) for that part of the value sought depending on the two terms affected with q and r. From whence the sum of any other two corresponding terms will be had, by barely substituting one letter, or value, for another: So that,
| 1 ⁄ n × | -log. (1 - x) | |
| -co-s. Q × M | + sin. Q × 2N | |
| -co-s. Q´ × M´ | + sin. Q´ × 2N´ | |
| -co-s. Q´´ × M´´ | + sin. Q´´ × 2N´´ | |
| -&c. | + &c. |
will truly express the sum of the series proposed to be determined; M, M´, M´´ &c. being the hyperbolical logarithms of 1 - 2αx + xx, 1 - 2βx + xx, 1 - 2γx + xx, &c. N, N´, N´´ &c. the arcs whose sines are x√(1 - αα) ⁄ √(1 - 2αx + xx), x√(1 - ββ) ⁄ √(1 - 2βx + xx), x√(1 - γγ) ⁄ √(1 - 2γx + xx), &c. and Q, Q´, Q´´, &c. the measures of the angles expressed by 360° ⁄ n × m, 2 × 360° ⁄ n × m, 3 × 360° ⁄ n × m, &c. And here it may not be amiss to take notice, that the series xm ⁄ m + xm ⁺ ⁿ ⁄ m + n + xm ⁺ ²ⁿ ⁄ m + 2n + &c. thus determined, is that expressing the fluent of xm ⁻ ¹ẋ ⁄ 1 - xⁿ; corresponding to one of the two famous Cotesian forms. From whence, and the reasoning above laid down, the fluent of the other form, xm ⁻ ¹ẋ ⁄ 1 + xⁿ, may be very readily deduced. For, since the series (xm ⁄ m - xm ⁺ ⁿ ⁄ m + n + xm ⁺ ²ⁿ ⁄ m + 2n - xm ⁺ ³ⁿ ⁄ m + 3n &c.) for this last fluent, is that which arises by changing the signs of the alternate terms of the former; the quantities p, q, r, &c. will here (agreeably to a preceding observation) be the roots of the equation zⁿ + 1 = 0; and, consequently, α, β, γ, δ, &c. the co-sines of the arcs 180° ⁄ n, 3 × 180° ⁄ n, 5 × 180° ⁄ n, &c. (as appears by the foregoing note). So that, making Q, Q´, Q´´, &c. equal, here, to the measures of the angles 180° ⁄ n × m, 3 × 180° ⁄ n × m, 5 × 180° ⁄ n × m, &c. the fluent sought will be expressed in the very same manner as in the preceding case; except that the first term, -log. (1 - x) (arising from the rational root p = 1) will here have no place.
After the same manner, with a small increase of trouble, the fluent of xm ⁻ ¹ẋ ⁄ 1 ± 2lxⁿ + x²ⁿ may be derived, m and n being any integers whatever. But I shall now put down one example, wherein the impossible quantities become exponents of the powers, in the terms where they are concerned.
The series here given is 1 - x + x² ⁄ 2 + x³ ⁄ 2.3 + x⁴ ⁄ 2.3.4 - x⁵ ⁄ 2.3.4.5, &c. = the number whose hyp. log. is -x, and it is required to find the sum of every nth term thereof, beginning at the first. Here the quantity sought will (according to the general rule) be truly defined by the nth part of the sum of all the numbers whose respective logarithms are -px, -qx, -rx, &c.; which numbers, if N be taken to denote the number whose hyp. log. = 1, will be truly expressed by N⁻px, N⁻qx, N⁻rx, &c. From whence, by writing for p, q, r, &c. their equals 1, α + √αα - 1, α - √αα - 1, β + √ββ - 1, β - √ββ - 1, &c. and putting α´ = √1 - αα, β´ = √1 - ββ, &c. we shall have 1 ⁄ n × (N⁻px + N⁻qx + N⁻rx), &c. = 1 ⁄ n into N⁻ˣ + N⁻ᵃˣ × (N⁻ᵃ´ˣ√⁻¹ + Nᵃ´ˣ√⁻¹) + N⁻ᵝˣ × (N⁻ᵝ´ˣ√⁻¹ + Nᵝ´ˣ√⁻¹) + &c. But N⁻ᵃ‘ˣ√⁻¹ + Nᵃ‘ˣ√⁻¹ is known to express the double of the co-sine of the arch whose measure (to the radius 1) is α´x. Therefore we have 1 ⁄ n into N⁻ˣ + N⁻ᵃˣ × 2 co-s. α´x + N⁻ᵝˣ × 2 co-s. β´x, &c. for the true sum, or value proposed to be determined.