will truly express the sum of the series proposed to be determined; _M_, _M´_, _M´´_ &c. being the hyperbolical logarithms of 1 - 2α_x_ + _xx_, 1 - 2β_x_ + _xx_, 1 - 2γ_x_ + _xx_, &c. _N_, _N´_, _N´´_ &c. the arcs whose sines are _x_√(1 - αα) ⁄ √(1 - 2α_x_ + _xx_), _x_√(1 - ββ) ⁄ √(1 - 2β_x_ + _xx_), _x_√(1 - γγ) ⁄ √(1 - 2γ_x_ + _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_ + 2_n_) + &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_ + 2_n_) - _xᵐ ⁺ ³ⁿ_ ⁄ (_m_ + 3_n_) &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 ± 2_lxⁿ_ + _x_²_ⁿ_) 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.