Make, now, _p_³ = 1, _q_³ = 1, and _r_³ = 1; that is, let _p_, _q_, and _r_, be the three roots of the cubic equation _z_³ = 1, or _z_³ - 1 = 0: then, seeing both the second and third terms of this equation are wanting, not only the sum of all the roots (_p_ + _q_ + _r_) but the sum of all their squares (_p_² + _q_² + _r_²) will vanish, or be equal to nothing (by common algebra), as they ought, to fulfil the conditions of the two first equations. Moreover, since _p_³ = 1, _q_³ = 1, and _r_³ = 1, it is also evident, that _p_⁴ + _q_⁴ + _r_⁴ (= _p_ + _q_ + _r_) = 0, _p_⁵ + _q_⁵ + _r_⁵ (= _p_² +_q_² + _r_²) = 0, _p_⁶ + _q_⁶ + _r_⁶ (= _p_³ + _q_³ + _r_³) = 3. Which equations being, in effect, nothing more than the first three repeated, the values of _p_, _q_, _r_, above assigned, equally fulfil the conditions of these also: so that the series arising from the addition of three assumed ones will agree, in every term, with _that_ whose sum is required: but those series’ (whereof the quantity in question is composed) having all of them the _same form_ and the _same coefficients_ with the original series _a_ + _bx_ + _cx_² + _dx_³, &c. (= _S_), their sums will therefore be truly obtained, by substituting _px_, _qx_, and _rx_, successively, for _x_, in the given value of _S_. And, by the very same reasoning, and the process above laid down, it is evident, that, if every _nᵗʰ_ term (instead of every third term) of the given series be taken, the values of _p_, _q_, _r_, _s_, &c. will then be the roots of the equation _zⁿ_ - 1 = 0[155]; and that, the sum of all the terms so taken, will be truly obtained by substituting _px_, _qx_, _rx_, _sx_, &c. successively for _x_, in the given value of _S_, and then dividing the sum of all the quantities thence arising by the given number _n_. The same method of solution holds equally, when, in taking every _n_ᵗʰ term of the series, the operation begins at some term after the first. For all the terms preceding _that_ may be transposed, and the whole equation divided by the power of _x_ in the first of the remaining terms; and then the sum of every _nᵗʰ_ term (beginning at the first) will be found by the preceding directions; which sum, multiplied by the power of _x_ that before divided, will evidently give the true value required to be determined. Thus, for example, let it be required to find the sum of every third term of the given series _a_ + _bx_ + _cx_² + _dx_³ + _ex_⁴, &c. (= _S_), beginning with _cx_². Then, by transposing the two first terms, and dividing the whole by _x_², we shall have _c_ + _dx_ + _ex_² + _fx_³, &c. = (_S_ - _a_ - _bx_) ⁄ (_xx_) (= _S´_). From whence having found the sum of every third term of the series _c_ + _dx_ + _ex_² + _fx_³, &c. beginning at the first (_c_), that sum, multiplied by _x_², will manifestly give the true value sought in the present case.