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_ + 2_n_) &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) + _x_² ⁄ (_m_ + 2) + _x_³ ⁄ (_m_ + 3), &c. = -(1 ⁄ _xᵐ_) × H. Log.(1 - _x_) - (_x_ + ½_x_², &c.) ⁄ _xᵐ_. In which value, let _px_, _qx_, _rx_, &c. be, successively, substituted for _x_ (according to prescript) neglecting intirely the terms (_x_ + ½_x_²) ⁄ _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_));