Elementary principles of the theories of electricity, heat, and molecular actions / Designed for the use of students in the University. By the Rev. R. Murphy... Part I. On electricity.

Murphy, Robert, 1806-1843.

Date:: 1833

Credit: Elementary principles of the theories of electricity, heat, and molecular actions / Designed for the use of students in the University. By the Rev. R. Murphy... Part I. On electricity. Source: Wellcome Collection.

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$and if we now omit the accents, we get the same expression as that already obtained for tm, which is therefore true for all values of m, from — 1 to +cc , provided t is between 0 and 1. Thus, let m = — we get 7= — ~ ^ Pq + P\ + A T &c- \ V t which may be confirmed by observing that by Prop, n. we have {l — 2h (l — 2t) + h' \ = P0 + Plh + PJr + &c. and then putting h — 1. Again, if we differentiate the general expression for tm, and then put m = 0, we get - log. (t) = (P0 + A) + i (A + A) + i (A + A) + &c. A simple numerical illustration may be given by putting t — T when we have P2n = coefficient of h2n in (l 4- by Prop. ii. 1.3.5...2n— 1 = (- l)n. ----, 2.4..6... 2 72 and P2n+1 = 0; 1 u = V(i) = 1 “ 1 . 1.3.5 + 2.4 2.4.6 4- &c. and - log. (T) = log. (2) = 1 2.3 2 • o + 9 1.3 4.5 2.4 &c. Prop. VII. To compare the indefinite integral of Pn of any order not higher than the nth, and commencing when t - 0 with its diffe¬ rential coefficient of the same order. If we denote the mth integral of Pn by the symbol ftm Pn i . 1 dn(tt')n then, since P„ —-.—--, n 1.2.3...w dtn$