vol/. LXXXI.] PHILOSOPHICAL TRANSACTIONS. 45 



tion all the terms become negative or affirmative, and from the latter they be- 

 come alternately negative and affirmative, then will a be less than the least root 

 of the given equation. If in the same manner, in the given equation for x be 

 substituted v -\- a and v — a, and the terms result as before, then will a be 

 greater than the greatest root, affirmative or negative, of the given equation. 



When the integral of an algebraical quantity, whose increments are finite, is 

 required ; first, by the method given in Medit. Analyt. investigate the integral in 

 finite terms, if it can be expressed by them ; but if not, reduce it into infinite 

 serieses of which the integral of each of the terms can be found, and also the 

 serieses for finding the integral contained between the 2 different given values of 

 the variable quantity may converge. Serieses of this kind have been given in the 

 Medit. Analyt. and innumerable of a like kind may be added for finding integrals 

 by converging serieses either ascending or descending, of which the given incre- 

 ments are either finite or evanescent. 



It may be observed, that generally the particular case of which the increments 

 are nascent or evanescent may be deduced from the general, in which the incre- 

 ments are finite ; and consequently in many cases the general will, mutatis mu- 

 tandis, correspond to the particular; e. g. 1. the integral cannot be expressed in 

 finite algebraical terms, when any factor in the denominator of the increment 

 has not a successive correspondent one ; which is analogous to the case of the 

 simple divisor in the denominator of a fluxion published in the Quadrature of 

 Curves. 2. Nor can it be expressed by the above-mentioned terms, when the 

 dimensions of the variable quantity in the denominator exceed its dimensions in 

 the numerator by unity, which corresponds to a similar case in fluxions first given 

 in Medit. Analyt. To these may several others be added. After several exam- 

 ples, it is added, it appears therefore, that a series will terminate equally by an 

 ascending or descending series ; and the end of the one ascending series is the 

 beginning of its correspondent descending one. 



It has been observed in the Medit. Analyt. that if some quantities contained in 

 the given irrational ones are much less or greater than the rest, it may be pre- 

 ferable in the former case to reduce them into serieses not proceeding according 

 to the dimensions of x, but according to the dimensions of those quantities ; and 

 in the latter case according to the reciprocal dimensions of them ; and particularly 

 so if the fluent or integral of the terms of the resulting serieses can be found in 

 finite terms, or by tables already calculated. From similar principles to those 

 before given may be found when the resulting series will converge, and when not. 

 This method will in many problems be useful, when the value of a near approxi- 

 mate is known. Of this case 'Dr. W. subjoins a few examples, of which some 

 have been already published in the Medit. Analyt. 



