132 PHILOSOPHICAL TRANSACTIONS. [aNNO 1777. 



and in like manner, the differences of the said 3d differences, or the 4 th dif- 

 ferences, of the original numbers ; and the 5th and 6th differences, and all 

 higher differences, of the same numbers, also form decreasing progressions. 

 Also let T be a quantity of any magnitude not greater than unity. Then on 

 these suppositions the value of the infinite series a — bx -j- cxx — dx^ -\- ear'* — 



fx^ -j- gx*^ — hx'' + &c. will be equal to the following differential series, viz. 

 lix p'.i'j D",r' d"'x* D'\r' D",r' . i 



\+x (1 + x)^ (l + .r)J {\+xy (1+x)' (1+x)" '^^•. vvucrc 



n' =: b — c, the 1st of the 1st order of differences ; 

 d" = A — 2c + d, the 1st of the 2d order of differences ; . , 



d'" = b — 3c -\- 3d — e, the 1st of the 3d order of differences ; 

 D'v = b — 4c -^ 6d — 4e -{-f, the 1st of the 4th order of differences; and so on. 



This theorem is the same, in substance, as the 7th prop, of Stirling's Me- 

 thodus Differentialis, first published in the year 1730; and it is the same, both 

 in form and substance, as corol. 4, pa. 65, of Simpson's Dissertations, pub- 

 lished in 1743. Mr. Maseres has not given the investigation of this series ; 

 but it is given by Stirling, and still in a way much neater and easier by Simpson ; 

 by both of whom also the method is applied to a series, the terms of which have 

 all the same sign, -F or — , as well as to the case above given, where the terms 

 have these signs alternately. 



Mr. Maseres remarks that the foregoing differential series will always con- 

 verge with a considerable degree of swiftness, so that 6 or 8 of its terms will 

 give the value of the whole, and consequently that of the original series to 

 which it is equal, exact to several places of figures, even in the most difficult 

 cases: for \{ x he = 1, which is its greatest possible magnitude, 1 -|- x will be 

 = 1 -|- 1 or 2, and consequently (l -|- x)", (1 -}- xf, (1 -f r)', (l -f xy, and 

 the following powers of 1 -f- x, will be equal to 4, 8, 16, 32, and the following 

 powers of 2 ; and the powers of the fraction will be equal to the powers 



-_,,.,. bx o'xx i)"x' d"'t* d"x' 



of 4. Therefore the series a — 



1 + X rr^'- TTl^' 1 + x)' TTTl' 



..,,.,,. , Ad' d" v'" v'^ dv 



&c. will m this case be=i:toa— - — T ~ J Tii ~ '32 ~ hi ~ "^^ 



terms of which decrease in a greater proportion than that of 1 to 2, because the ' 

 numerators a, b. d', d ', d", d'% d'', &c. form a decreasing progression, and the 

 denominators increase in the proportion of 2 to 1 . 



Mr. M. gives some examples of the use of this theorem, in the numeral cal- 

 culation of the series for the length of a circular arc that is expressed by a series 

 in terms of its tangent ; and also of the series expressing the time of a body 

 descending by gravity down a circular arc. Which examples it would be useless 

 to repeat in this place. 



