3.44 
Here a~i, d 1 -. 
d 
A 
|V_ 
12, d ,11 =6, d iy —o 
J 7/-2 
L G E 
there¬ 
fore the fum = n 4- —-X7+ ?! -~ 
2 
BRA. 
Co a. 3. If the increment be.'fxw-i' r > w +2r.m+37’. 
X 12- 
to 72- 
n — 1 77 —2 n —3 
X 6: 
7?‘‘ + 272 , +22’‘ 
H-i) 
when H is invariable, the integral is —x 
nr 
.... ? 72 +?z—i.rrfcC. 
„ 4 r , 4 If d be the confiant increment, and m the number of 
Sometimes the fum may be more readily obtained by timcs it has been taken the j me a ra l is wA±C. To find 
beginning the (cries with one or more cyphers; thus, to 
find the fum of n terms of the feries i 3 + 2 3 4 - 3 :i + &C. 
tjike ?/+1 terms of the feries o+i 3 4-2 3 +3 3 +&C. 
o, 1, 3 , 27, 64 
*> 7 . * 9 > 37 
t, 12, 18 
6 , 6 
o 
Here a— o, d'—x, d %t —6, d i 1 ‘=6; and the fum of n-A-i 
72+1 .n.n —1 .n —2 
. —n , n 4 -1 .n. n — 1 
terms is 72+1.-4- - -— X 
2 j.2.3 
X 6 
72+1 .72,. 2 4-7; + i .n ■ 
72- 
_ n - 2 - 3-4 
1.4+22+1.72 . n —1 .72- 
72+ 1.72.2+472-4+7i 2 - 
« . 72+ . 
The differential method may alfo he applied to the in¬ 
terpolation of feries, and the quadrature of curves. 
On the Method of Increments. 
Any variable quantity is called an integral. The magni¬ 
tude by which it is increafed at one Hep is called the incre¬ 
ment. Thus J+2+3.+ 772— 77 ;.and the mag- 
2 
nitude by which it increafes at one Hep is 772+1, which is 
772+1 
called the increment of the integral m. 
When the 
quantity decreafes, the increment becomes negative. 
If two quantities begin to increafe together, and their 
correfponding increments be always in the fame ratio, 
their integrals, or the whole quantities generated, will be 
in that ratio. Let the correfponding increments be 
A, B, C, Sec. and a, b, c, See. and let A : a :: B : b :: C: c See. 
::m: n) then - 4 +/ 1 +C+ &c. : <2+i+c+ &c. :: 772: ?2. 
Cor. When 772=77, or the increments are equal, the in¬ 
tegrals are alfo equal. 
If an integral be reprefented by the product of quanti¬ 
ties in arithmetical progreffion, as 772.? 72 +r.772+27-.777+37-.... 
w + 77—1. r, where r is confiant, and m is increafed at eve¬ 
ry Hep by r, the increment of this integral is !rX*+r. 
7 / 2 + 2 /-. 777+37-.... 777 + 77 —1. r. 
The firft value of the integral is 7?2.772+r. 772 + 27 -.7?*+3r 
... 772 + 72-1 .r, and the fucceeding value is 7s4-r. 
;ti+ 27 -.?7z+3r. 772 + 47 -. 772+727-, whofe difference, that is, 
the increment of the integral, is nr. 777 + 27 . 772 + 3 ^.. 
m+ 72 — 1 .r. 
Cor. i. Since an invariable quantity C has no incre¬ 
ment, the increment of m.m-\-r.m-\-zr.m-) r T,r....m-\-n — 1 . r 
+ C is alfo 727- X 772 +r.772 + 27 -. 772 + 37 - .... 772 + 72 -1 . 7\ 
Cor. 2. Hence if the increment of an integral be re- 
therefore the integral of any increment, let the increment 
be reduced to the piodufts of arithmetical progreffionals 
w hofe common difference is the quantity by which the va¬ 
riable magnitude is increafed at every Hep, and the inte¬ 
gral of each increment will be found by multiplying it by 
the preceding term in the progreflion, and dividing it by 
the number of terms thus increafed, and by the common 
difference. 
The conHant quantity which is to be added to or fub- 
trafted from this refult, in order to obtain the correct inte¬ 
gral, muH be determined by the nature of the quefiion; 
thus, when x, the integral obtained by the rule, is a, fup- 
pofe the true integral is known to be b ; then fince .v+C is 
in all cafes the integral, 72+C— b, or C—b — a; therefore, 
the con-eft integral is x-\-b — a. 
Ex. i. To find the fum of ?2 terms of the feries 1 + 2 + 
3 + &c. 
The ?2th term is n, and the increment of the fum is tz+i, 
w hofe integral according to the rule is ” And this 
is the correft integral, becaufe when 72=1, the firm is 
1.2 . , 
-=1, as it ought to be. 
Ex. 2, What is the?2th term of the feries 7, 9,11, &c- 
Here the increment is 2, and the integral 272+C, but 
when 72 =i, the integral is 7, therefore 2+C=7, and 
C— 5 ; therefore the correct integral is 272+5, the77th term 
of the propofed feries. 
Ex. 3. To find the fum of 72 terms of the feries T+ 2 ,1 4 '' 
3 2 + &c. __ 
The increment of this fum is «+i !’=??.77+1+77+1, 
, , . , r . n - 1 . 77 . 77+1 72 . 72+1 
and the integral or fum is--= 
__ 3. 2 
277 +J . 72 . 72+1 
which needs no correftion. 
2 -3 
Ex. 4 . To find the fum of 72 terms of the feries 1 '+3’+ 
S 2 + &c. 
The increment of the fum is 222 + 1 ] a = 472 ! +472 +i = 
4-77-1.77.77+1 
+72; which 
472 . 7 Z+I+I, whofe integral is 
4 
needs no correftion. 
Ex. 5 . To find the fum of n terms of the feries i 3 + 2 ®+ 
3 3 +&c. 
The increment of the fum is «+i ) 3 - 
:7i. 7z+i. 72+2 +72+i, whofe integral is 
72 . 72+1 
- 72 3 + 3 72 2 +3 77 + l, 
72-1 . 72 . 72+ I .72 + 2 
, the fum required; which needs no 
2 2 
correftion. 
The following table of figurate numbers is formed by 
making the 77th term of each fucceeding rank equal to the 
fum of n terms of the preceding : 
prefented by nr y,m-fr.m-\-ir.m-\-^r . 772+ n —1 .r, 
the integral is 772. 77i+r.??2+27-. 772 + 37 -. 772+72— 1 . r±C ; 
where the invariable quantity C mult be determined by the 
nature of the equation. 
3 
iH Order, 1, 
i, 
i> 
1 
2d 1, 
2) 
3 . 
4 > 
5 ? 
6 
3 ^ O 
3 » 
6, 
10, 
i 5 > 21 
4th 1, 
4 > 
10 , 
20, 
35 > 
5° 
5 th 1 > 
&c. 
S> 
J 5 > 
drc. 
35 > 
70, I26 
Ex. 6. To find the fum of 72 terms of the 772th order of 
figurate numbers. 
The 
