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24p1

(p − 1)(2p — 1)(3p — 1) (4 p − 1) Do — &c.

p-1

120p

D; = D; — P — ' D2 + (p − 1)(11p-7) D2

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12p1

DA

(p-1)(2p-1)(5p-3) Do+ &c.
5
12p

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

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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.

=

An om

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:

8

* Dru=Sm

Anon+m

\n+m-1

(Dx)n+m-1.dn+m-1 u.

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

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this series converges very rapidly.

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

INVERSE METHOD OF DIFFERENCES.

(19.) Integration of algebraical functions.

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Σ ̧a.u ̧=aΣ ̧ux; if a is either independent of a, or such

a function of x, that a=a+1°

Σ, {1u ̧ + 2u ̧ + &c, +"u ̧} = Σ ̧1u2+ Σ 3u„ + &c. + Σ„”u«
Σ.(κ.Δ.) =10 - Συ+1.Δεν

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

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Anx2 + An-122 – 1 + &c. + A,∞ + A

2

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n

1

B+B1.u2+ B2. U ̧· U2+1+&c. + B2. u.u2+1... Uz + n − 19 developing the latter quantity in powers of x, and equating coefficients.

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(Tr. L. App. 368—72; L. C. D. 943-54; L. D. C. 516-23.)

(21.) The numerical coefficients of a-1, -3, &c. are the numbers of Bernoulli; let them be represented by 62, 64, &c. respectively, then

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