Method of correfpondent Values, &c. 
3, n 
n — I 
n 
n -3 n — 4 . 
l ll 
, which are 
numbers which are 
. a 3 4 5 
5, & c. ; and the fum of the contents of every n of the former 
will be lefs than the fum of the contents of every n latter 
numbers by 1 . 2 * 3 . 4 . . n — 1. 
1 2. The method given in Art. 4. which I name a method of 
correfpondent values, eafily deduces and demonftrates the pre- 
ceding equations, which cannot, without much difficulty, be 
done by the preceding method of differences ; the method of 
correfpondent values is much preferable to the method of dif- 
ferences, both for the facility of its deduction, and the gene- 
rality of its refolution : for inftance, from this method very 
eafily can be deduced, &c. the fubfequent and other fimilar 
equations. 
n — 1 
Ex. i. Sn=*nSn — 1 
£*=#Siq=S nearly. 
n . • — - S« — - z 4- # • 
S n 
Ex. 2 . S n + m=z 
m-{- n • m 4 rn — I . m- 4 -n — 2 . . . m+ 2 
X S 71 
n~ 1 
m+i 
x A x 
I m 4-2. 
x Sn — 2 + 
72 ’ 
• 3 • • *-1 
lxBx m+2 
2 m + 3 
m 4 - 4 
S/z — 3 
3 
X ! 
xS«- 44- —i x D x AZL 5 * x S » — 5 - &c. nearly, where the 
^4-4 4 ^ + 5 
letters A, B, C, D, &c. denote the preceding co-efficients, and 
the converging feries is the fame as in the preceding example. 
Ex* 3. Let the converging feries be of the formula ax + i>x* 
cx 5 4 -dx 1 4- &c. ; then will Sn == zn - 2 Sn — 1 — zn — 1 x ! 
2»— 4-r« ■ . 2n-*-2 2 # — 6 o " ' ‘ ‘ 2K—2 
Sn — 2 4- 2« 
2 w— 2 
I x X 
S/Z — 3 — 2» — I 
2n ^ x n - 44-&C. nearly, of which the general term is 
2/2 — i 
4 
2 n — 2 
2n ~3 . . x ilzl 7 x S^ 7 . 
2 3 
Vol. LXXIX. 
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