[ 405 * ] 



(if the abfolute value of the leading term be unvaried) the ferics 

 will converge fafter. If then the given number whofe lognri'thm 

 is fought, be denoted by a fimple quantity .v, the lower the 

 place which the fimple quantity occupies among the fadors, or 

 the fewer refidual fadors there are, the fafter will the feries con- 

 verge. — Now whatever compound quantity be fubftiluled for this 

 fimple one, the efFed ^"ill be the fame, viz. that the convergency 

 of the feries is quicker or flower, the lower or higher the place 

 which the given number occupies in the rank of fadors. 



But fince it is ufeful that the difference of the greateft and leaft 

 fador fhould be as fmall as poflible (as follows from what has 

 been obferved before) the advantage of a fwifter convergency, 

 will, generally fpeaking, be in a lefs degree in this cafe, than 

 the fimilar advantage which arifes from multiplying the fecond 

 term of each fador by an aliquot fradion, or (which amounts to 

 the fame thing) the leading term of each fador by the reciprocal 

 integer. 



From what has been faid it follows, that if, of the firft and 

 fecond, fourth and fifth terms of an arithmetical progreffion, the 

 logarithms of any three be given,, the logarithm of the remaining 

 term may be found. If the common difference of the progref- 

 fion be unity, — , will be equal to an aliquot fradion, whofe 



denominator 



