Ꭰ 24p1 (p − 1)(2p — 1)(3p — 1) (4 p − 1) Do — &c. p-1 120p D; = D; — P — ' D2 + (p − 1)(11p-7) D2 12p1 DA (p-1)(2p-1)(5p-3) Do+ &c. these differences, which are given as far as the 5th order, are quite sufficient for the calculation of tables. 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. (18.) If the log. sines be calculated for large intervals, as for every 10°, the differences for every degree may be thus found: this series converges very rapidly. (Enc. Met. Trig. 194-209; L. C. D. 893—6.) INVERSE METHOD OF DIFFERENCES. (19.) Integration of algebraical functions. 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« in which the numerator is at least two dimensions lower than the denominator, may be decomposed into integrable fractions by assuming Anx2 + An-122 – 1 + &c. + A,∞ + A 2 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. (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 |