And here it may be worth while to observe, that all the terms preceding _that_ at which the operation (in any case) begins, may (provided they exceed not in number the given interval _n_) be intirely disregarded, as having no effect at all in the result. For if in that part ((-_a_ - _bx_) ⁄ _xx_) of the value of _S´_, above exhibited, in which the first terms, _a_ and _bx_, enter, there be substituted _px_, _qx_, _rx_, successively, for _x_ (according to the _prescript_) the sum of the quantities thence arising will be - _a_ ⁄ (_p_²_x_²) - _a_ ⁄ (_q_²_x_²) - _a_ ⁄ (_r_²_x_²) - _b_ ⁄ _px_ - _b_ ⁄ _qx_ - _b_ ⁄ _rx_ which, because _p_³ = 1, _q_³ = 1, &c. (or _p_² = 1 ⁄ _p_, _q_² = 1 ⁄ _q_, &c.) may be expressed thus; - _a_ ⁄ _xx_ × (_p_ + _q_ + _r_) - _b_ ⁄ _x_ × (_p_² + _q_² + _r_²) But, that _p_ + _q_ + _r_ = 0, and _p_² + _q_² + _r_² = 0, hath been already shewn; whence the truth of the general observation is manifest. Hence it also appears, that the method of solution above delivered, is not only general, but includes this singular beauty and advantage, that in all series’ whatever, whereof the terms are to be taken according to the same assigned order, the quantities (_p_, _q_, _r_, &c.), whereby the resolution is performed, will remain invariably the same. The greater part of these quantities are indeed _imaginary_ ones; and so likewise will the quantities be that result from them, when substitution is made in the given expression for the value of _S_. But by adding, as is usual in like cases, every two corresponding values, so resulting together, all marks of _impossibility_ will disappear. If, in the series to be summed, the alternate terms (_viz._ the 2d, 4th, 6th, _&c._) should be required to be taken under signs contrary to what they have in the original series given; the reasoning and result will be no-ways different; only, instead of making _p_³ + _q_³ + _r_³ (or _pⁿ_ + _qⁿ_ + _rⁿ_, &c.) = +3 (or +_n_), the same quantity must, here, be made = -3 (or -_n_). From whence, _pⁿ_ being = -1, _qⁿ_ = -1, &c. the values of _p_, _q_, _r_, &c. will, in this case, be the roots of the equation _zⁿ_ + 1 = 0.