or Integrals the Fluents

FLUENTS, OR INTEGRALS.

THE fluents of such expressions, as are the most likely to occur in the solution of physical problems, may be very conveniently arranged in the form of a TABLE; the principal materials of which will be extracted from Meier Hirsch's Integraltaldn. 4. Berlin, 1810. It might have been somewhat en larged by additional matter that may be found in the earlier publications of our countrymen Waring and Lander!, which have been particularly consulted on the occasion; but Waring's improvements relate most commonly to cases so complicated, as seldom to be applicable to practical purposes; and Landen's theorems, though incomparably more distinct and better arranged than Waring's, tend rather to the investigation of some elegant analogies, than to the facilitation of actual computations. Some of these, however, will be briefly noticed, and an improve ment in the mode of notation will be attempted, which, if universally adopted, would tend to save much unnecessary circumlocution in the enunciation of many general theorems.

I. The

earlier letters of the alphabet, as far as q, and sometimes r, are commonly employed to denote constant quantities; the subsequent letters generally for quantities considered as variable. They are here employed as relating indifferently to quantities posi tive or negative, and to numbers whole or fractional ; except 'when they are used as indices exponents.

2. The Italic character is employed, in preference to others, for denoting quantities in general, the Ro man for characteristic marks, as d for a fluxion, or differential, sin, cos, or f, c, for sine and cosine; and hl for hyperbolic logarithm. The long

however, not being otherwise used, serves very con veniently as a characteristic, to denote a fluent.

8. When the Italic letters m, n, p, q, r, or any others, are employed as indices, they are to be here understood as denoting any numbers without limita tion ; the Roman small letters, m, n, will be applied to whole numbers only, excluding fractions, but either positive or negative, or 0 ; the small Italic Ca pitals M, N, to positive numbers, whether whole or fractional, excluding negative numbers only ; and the small Roman Capitals M, N, to positive integers only, including however 0.

4. The characteristic E implies the sum of a finite number of terms, derived from all the possible vari ations of a quantity, which is here denoted by a small letter of the Greek alphabet.

5. A comma, in an index, denotes or.

6. The fluents, indicated by the table, are to be understood as corresponding equally to any particu lar values of the quantities concerned; so that, in order to obtain the expression of the definite quan tity required by the conditions of any problem, we must always take the difference of the two values found by substituting two values of the elementary variable quantities ; and this rule being general, it supersedes the necessity of introducing a constant correction of the fluent in each particular case.

7. Particular values of fluents, limited on both sides, are distinguished by accents,/ ' .