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_ + _qrx_² = -_A_ × log. (1 - 2α_x_ + _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 - 2α_x_ + _xx_) = -(2√(-1) × √(1 - αα) × _ẋ_) ⁄ (1 - 2αx + xx); where (√(1 - αα) × ẋ) ⁄ (1 - 2α_x_ + _xx_) expresseth the fluxion of a circular arch (_N_) whose radius is 1, and sine = (√(1 - αα) × _ẋ_) ⁄ (1 - 2α_x_ + _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_ × 2_N_) 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, { -log. (1 - _x_) { -co-s. _Q_ × _M_ + sin. _Q_ × 2_N_ (1 ⁄ _n_) × { -co-s. _Q´_ × _M´_ + sin. _Q´_ × 2_N´_ { -co-s. _Q´´_ × _M´´_ + sin. _Q´´_ × 2_N´´_ { -&c. + &c.