566 PHILOSOPHICAL TRANSACTIONS. [aNNO I789. 



4. In the same book, is assumed a series ax" + hx'^^' + car^+*' + ctr'+3' -f- &c. 

 of a more general formula than the preceding, and in it for x substituted 

 «, |3, y, S, &c., m ; and sa, s^, sy, s^, &c. ; s^ for the resulting sums, and 

 thence deduced sm = 



mr "K m' — i2' . m' — Y . m' —^ . 8cc . m" x m' ~- u.' . m' — y" .m' — ^ . &cc. 



"«•• X «• — /8' . »' — V' • «' - ^' . &C. ^ ^'^ "T" ^r ^ ^. __ ' . fc' _ y« . ;3' _ ^* . &c." X Sf3 -|- 



m" X m' — »' >fn' — fi' . m' — ^' . &c. j^ »i' x w* — «' . »w' — ^' . m' — y* x &c. 



~/ X V' — «' . V' - /8' . V' — ^' . &C. ^ ^'J' + ^•- X «5" - «• . ^■'' - ^' . J'' _ y' ."&~ 



X s^ + &c. nearly. 



Cor. If for r and * be assumed respectively 1, the series becomes ax + ^^' + 

 car^ + &c. of the same formula as the preceding : if r = O and * = 1, the series 

 becomes a -\- bx -^ cx^ -{• &c. The latter case will be the same as the former, 

 when one of the quantities «, substituted for x and its correspondent sum sx, 

 both become = 0, and the equation deduced in both cases the same. 



5. Ifir, ^, 0-, &c. respectively denote r,r +p3 r-\-2p, . ,r •{- (n — l)p, r + (n — 

 l)p, and T = r + np ; and s, si , s2, s3, . . s{n — 2), s(n — l), be the sums either 

 resulting from the series ax + bx"^ + ^^^ + ^c. or the series a -\- ax -\- bx^ -{- 

 cx^ + &c., which respectively correspond to the values r, r -\- p, r -\- Ip, &c. of 

 X ; and sw the sum of the same series which corresponds to the value r -\- np o{ 

 X ; then will sn =. ns{n — l) — n . "-^ s(n — 2) + n . -^^ . ^-^ s(n — 3) 



"- • • i^ • '~h~ s2 + nsl + s nearly ; this equation differs from the preceding 

 by the last term s not vanishing ; in the preceding case s became = 0, for it was 

 the sum of the series ax -\- bx* ■\- ca^ -\- &c. which corresponded to a? = 0. 



6. From the Meditationes it appears that r"" — - w X (r + pY + n . 



"-=li (r + 2pY — n . "^^ . ^^ (r + 3pY + &c. to the end of the series = O, 

 if m is less than n, and m and n are whole numbers ; but if ot = w, then it 

 will = + 1.2. 3. 4. ..72— 1. np"" ; whence it is manifest, that for the n 

 first terms of the series a -^ ax -\- bx^ -^ &c. the equations are true ; and for 

 the n — 1 first terms of the series ax + bx"^ + cx^ + &c. and in the successive 

 term of both the serieses, they will err by a quantity nearly = + 1.2.3.. 

 n X p"* X r-" X co-efficient of the term ; and the errors of every subsequent 



term 3^4+" , will be nearly as + w . — — . — g— . — j— . . . ^ — X p X 



r- X co-efficient of the term 0?*+", if for r, r + j&, r -J- 2/), &c. be substituted 

 1, 1+5, 1+|, &c. 



n — 1 



7. Let the preceding equation sw = ns(n — l) — w . -— - s{n — 2) + w . 

 -^•-/ . s(n-3) -hc. = nX log.(r-.j&) - n . -^-log. (r - 2p) + 



n . ^-^ . ^— log. (r — 3p) + &c. but log. r — n X log. {r —p) -\- n. ^-^ 

 log. (r - 2j&) - n .-"^ . ^-1 X log. (r - 3/j) + &c. = log. 



