The series propounded, whose sum (_S_) is supposed to be given (either in algebraic terms, or by the measures of angles and ratio’s, _&c._) I shall here represent by _a_ + _bx_ + _cx_² + _dx_³ + _ex_⁴, &c. and shall first give the solution of that case, where every third term is required to be taken, or where the series to be summed is _a_ + _dx_³ + _gx_⁶ + _kx_⁶, &c. By means whereof, the general method of proceeding, and the resolution of every other case, will appear evident. Here, then, every _third_ term being required to be taken, let the series (_a_ + _dx_³ + _gx_⁶, &c.), whose value is sought, be conceived to be composed of _three_ others. ⅓ × (_a_ + _b_ × (_px_) + _c_ × (_px_)² + _d_ × (_px_)³ + _e_ × (_px_)⁴, &c.) ⅓ × (_a_ + _b_ × (_qx_) + _c_ × (_qx_)² + _d_ × (_qx_)³ + _e_ × (_qx_)⁴, &c.) ⅓ × (_a_ + _b_ × (_rx_) + _c_ × (_rx_)² + _d_ × (_rx_)³ + _e_ × (_rx_)⁴, &c.) having all the _same form_, and the _same coefficients_ with the series first proposed, and wherein the converging quantities _px_, _qx_, _rx_, are also in a determinate (tho’ yet unknown) ratio to the original converging quantity _x_. Now, in order to determine the quantities of these ratios, or the values of _p_, _q_, and _r_, let the terms containing the same powers of _x_, in the two equal values, be equated in the common way: So shall, ⅓ _b_ × _px_ + ⅓ _b_ × _qx_ + ⅓ _b_ × _rx_ = 0 ⅓ _c_ × _p_²_x_² + ⅓ _c_ × _q_²_x_² + ⅓ _c_ × _r_²_x_² = 0 ⅓ _d_ × _p_³_x_³ + ⅓ _d_ × _q_³_x_³ + ⅓ _d_ × _r_³_x_³ = _dx_³ ⅓ _e_ × _p_⁴_x_⁴ + ⅓ _e_ × _q_⁴_x_⁴ + ⅓ _e_ × _r_⁴_x_⁴ = 0 &c. And consequently, _p_ + _q_ + _r_ = 0 _p_² + _q_² + _r_² = 0 _p_³ + _q_³ + _r_³ = 3 _p_⁴ + _q_⁴ + _r_⁴ = 0, &c.