+

12p

(p-1)(7p— 5) 'DI - &c.

2p%

p-1 (p-1)(2p-1)
D, D
D2+

)
р
2p

брз
(p-1)(2p-1)(3p – 1)

24p
(p-1)(2p— 1)(3p— 1)(4p— 1) 05 - &c.

120p

(p-1)(11p-7) D:D:- DID? +

D

12p*
(p-1)(2p – 1)(5p --3) D+&c.

3(p-1)
=

D1+

4pø.
2(p-1)
= D4

D + &c.
D

- &c. these differences, which are given as far as the 5th order, are quite sufficient for the calculation of tables.

ANON (17.) Du= dru.(Dx)"

1.2...n

An On+1 +

dn+u,(D2)*+1 + &c.

1.2... (n+1) In this series, the values of du, &c. and Dx must be determined in each particular case: for intervals of 1',

Dx=0,000290888208665721596, &c.

Anom
The values of

will be the same in

every case; these 1.2...m for all values of m and n, from 1 to 12 inclusive, are given in the annexed table:

у

Aron + m-1 y DuurSm (Dx)"+m-1.drum-lu.

n+m-1

(18.) If the log. sinės be calculated for large intervals, as for every 10°, the differences for every degree may be thus found :

(sin (x + DX) - sin Dzlog sin x=2

sin (+ Dx) + sin x
(sin (x + Dx)-

- sinx
+ }

sin (v + Dx) + sin x D, sin a

De sin x = 2

+ }
12 sin x +D, sin a 2 sin x +D, sin
De log, sin æ= -2

(sin Dx)
(cos Dx) + cos (2x + DX)
(sin Dx)?

+ &c.; (cos Dx) + cos (2x + Dx) this series converges very rapidly.

(Enc. Met. Trig. 194—209; L. C. D. 893-6.

+&c.},

:* { cos Day 7,cos (2x + Dau)

)+&c.}

3

+

+}(

)

&c.}

INVERSE METHOD OF DIFFERENCES.

(19.) Integration of algebraical functions.

ExAxUz=Uz + const.

Exa.uz=aXxUz; if a is either independent of w, or such a function of x, that a=Qz+1

2:{u+ *ux + &c. + "u,} =,'Uz + $,u, + &c. + ], "UF.
Ex(u,.A.0.)=U70.-8.04+1-4,U..
Let u,=a + bx, then

U7-1•U•Uz+1...Ur+n-1
2.(;• Ug+2 ...Ux+n-1)

(n+1)
1

-1
Uz•Uz +1...
...Urtn-1

(n-1)b.up.u,+1.1,+n-3

4,2" + A- 20"-1 + &c. + 4,0 + A The fraction

U, U

Ux+n+1

;

# + 1

in which the numerator is at least two dimensions lower than the denominator, may be decomposed into integrable fractions by assuming

A + Ar-12-1+ &c. + A,« + A =B+B .uz + B.U,.Ur+1+&c. + B^.U.Uz+1...Ug+n-19 developing the latter quantity in powers of X, and equating coefficients.

(20.) 8,1= x + const.

9x= x(x-1)+const.
Ex=*.}.(— 1)(2x – 1) + const.
2,28=+*(x 1)+ const.
&c. =

&c.

m

-1

=

com

m - 3

+

m...

11

2- 13

1 E.com

12m + m +1

2.3 2 1 m(m-1)(m-2)

1 m(m-1)...(m-4)

+ 2.3.5 2 3 . 4

2.3.7 2 3

6 1 m(m-1)...(m–6)

5 m(m-1)...(m—8)gme 2.3.5 2 3

8
2.3.11 2 3

10
691 m...(m-10)

7 (m – 12) com

+ 2.3.5.7.13 2... 12

2.3 2... 14 3617 m (m— 1)... (m – 14)

cm – 15 + &c. + const. a 2.3.5.17 2 3

16 (Tr. L. App. 368–79; L. C. D. 943–54; L. D. C. 51693.) (21.) The numerical coefficients of 2-1, 2-, &c. are the numbers of Bernoulli ; let them be represented by 62, 64, &c. respectively, then

0=}-} +62)
0= }-} +962 +6

+

3

.

or

m(m - 1)...(m - 4) m(m — 1)... (m–6)
Go +

68 + &c. 2 3

6
2

-8
(Moivre, Miscell. Anal. Supp. p. 6.)
tem

t 6em is the coefficient of

in the expansion of 1.2.3...2m

-1 2 m

2m 62m=(-1)^-1

(22m –

12m – 1) 22m (22m — 1)

1

+ (32m – 1 – &c.) - &c.}. B 1 = d.

(TT. L. App. 408.) *

2m

2m - 1

{1

2m

(22.) Integration of exponential functions.

at Eram =

+ const.

a

sin (w++) @ Ercos 0=

+ const. 2 sin 10

cos (x + 1) sin =

+ const. 2 sine

sin (a ++4.6) Ecos (a + b) 0 =

+ const. 2 sin 160

cos (a + x +4.60 E, sin (a + bx)=

+ const. 2 sin 460