VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 5QQ 



w, &c. ; then find the sum of the series, first, from the hypothesis that one of 

 them z is only variable, which, properly corrected, let be a ; in the quantity a 

 suppose all the quantities invariable but some other v, and find the sum of the 

 series thence resulting, which let be b, and so on ; and the sum of the series 

 will be deduced. 



Exam. Let the term be — — — ; the dimensions of zandr. he. 



z.z + lXl>.t>+l.t' + 2 



in the denominator must be at least greater than its dimensions in the numerator 

 by a quantity greater than the number of the quantities z, v, &c. which proceed 

 in infinitum increased by unity. First, suppose z only variable, and the sum of 

 the infinite series resulting will be — — = a ; then suppose v only 



variable, and the sum resulting will be — — — = b, which is the sum re- 

 quired. If it be supposed, that the quantities z and v, &c. in the same term shall 

 never have the same values, then suppose z and v always to have the same values, 



and the general term ———-}———— becomes -___I_^ G f which 



let the sum be v, then will b — v be the sum required. On this and some other 

 subjects more have been given in the Meditationes. 



]4. If the sum of the series cannot be found in finite terms, and it is necessary 

 to recur to infinite series ; it is observed in the Meditationes to be generally 

 necessary to add so many terms together, that the distance from the first term of 

 the series may considerably exceed the greatest root of an equation resulting from 

 the general term made = O ; and afterwards a series more converging may com- 

 monly be deduced from the fluents of fluxions resulting from neglecting all but 

 the greatest quantities in the general terms resulting ; and by other different 

 methods. Mr. Nicholas Bernoulli and Mr. Monmort investigated the sum of 

 the series (p) a + b ? + cr~ + &c. by a series (a) ^-j -f JT^Ty + 7T^7)1 + 



- 1- &c. ; where d', d' d'", &c. denote the successive differences of the 



(i - »•)•» 



terms a, b, c, d, &c. If r be negative, the denominators become 1+r, (i-j-r) 2 , 

 (1 + r)\ &c. 



It has been observed, in the Meditationes, that in swift converging series the 

 series p will converge more swiftly than the series q ; in series converging accord- 

 ing to a geometrical ratio, sometimes the one will converge more swift, and some- 

 times the other. In other series, which converge more slowly, where most com- 

 monly r nearly = 1 , it cannot in general be said, which of the serieses will con- 

 verge the swiftest. The preceding remark, viz. the addition of the first terms of 

 the series, is necessary in most cases of finding the sums by serieses of this kind. 



It is not unworthy of observation, that in almost all cases of infinite series, the 

 convergency depends on the roots of the given equations ; which remark was first 



