Method of correfpondlng Values , &c. iyn 
6. From the Meditationes it appears, that — n X r-±zp* + 
n . rz±z zp - n . . r ^3p +&c. to the end of the 
2 2 3 ~ 
fei 'ies = o, if m is lefs than n 9 and m and n are whole num- 
bers ; but if m~n, then it will « r . 2 . 3 . 4 . . . « — i % np m \ 
whence it is manifeft, that for the « firft terms of the feries 
A + ax + bx z 4 -&cc. the equations are true; and for the n - 1 
firft terms of the feries ax + bx 1 -f cx 3 + &c. and in the fucceffive 
term of both the feriefes they will err by a quantity nearly 
= — 1 . 2. 3 .. n x p n x r~ n x co-efficient of the term ; and the 
errors of every fubfequent term (jv 4+») will be nearly as 
—m . * . • * . * — ; x p n x r— n x co-efficient of 
2 3 4 £ 1 
the term **+”, if for r, r 4- p, r-\-2p, &c. be fubftituted i, 
i + — , i + 2 -, See.. 
r r 
j. Let the preceding equation S n = — i - » S« — 2 
-p Tl • 
72 — I 77 — 2 
. S/z — 3 — &c. ~ n x log, r—p—n . 
72 — 1 
2 
log. r — zp n 
n — 1 ?? — 2 
log, r — 3 p + &c. =: 
r xr--,/ xr-_y = log, K, where s, s', s", &c. de- 
r — p ‘ ~ r — Sp * xr — 5/ X &c„ 
note the co-efficients of the alternate terms of the binomial 
, • n — I / w— 1 72 — 2 72 — q 0 1 
theorem, w#- s~n e - — , s =n . — . . — 2 an d 
2 2 3 4 
/ = «, t' ~n . 2^ • 2 Z _3 &c. the co-efficients of the remain- 
2 3 
1 1 ■ — ■ j 1 s 
ing alternate terms ; the numerator r x r — 2p x r — ^p x 
r — 6 p x &c. = (if N = 2”“ x ) r N — P^r N— 1 4- Q^V N -"' 2 — Ry>V N — 3 
. . . L^*” 1 x ; and the denominator 
r ~p* xr - ^p 1 x r - 5/ x &c. = r N — Ppr **” 1 + Q/>V N ~ 2 — 
