APPROXIMATION.
All that precedes is found in elementary treatises, with the exception of a few terms of the last series: we now come to matter which has been hitherto only the property of the well-read mathematician, but which well deserves to be made as common as Taylor's Theorem. We refer to Arbogast's metAod of derivations. (Aesooasr, in Moo. Div.] Few, even among mathematicians, are aware of the power of this process, which may perhaps arise from their taking Lacroix. account of it, instead of consulting the work of Arbogast himself : the former has only exhibited it to show that it may be reduced to processes of the differential calculus; and even the latter has so loaded his method with heavy applications, that he has concealed much of its beauty and simplicity.
The foundation of Arlxvast's methods is a contrivance for expediting the expansion of ta (a + + ex= + . . . .) into a series of the form + + art + The process by which D is formed from A, c from n, fie.. is uniform, and is called derivation ; and A being tier, n may be called boa, c may be called nrea, or Dlcpa, and so on. Hence 6 ought to be called DO, c ought to be Oa, and so on. This notation is not precisely that of Arbogast, but will do for our purpose. For more detail; see the Differential Calculus (' Library of Useful Knowledge '), pp. 328-334.
If, for a moment, we write the expansion thus + + &c.) &c.
and if we differentiate both sides with respect to a., x and all the other coefficients remaining constant, we have dk, op' + &c.) = x ' which shows that a. cannot enter any coefficient preceding A., or da. +1 The first side of this is the same series, whatever letter a., was made to vary ; the second side is therefore always the same series ; whence we collect that da.,.„, : da . does not alter with the value of et, being always the coefficient of x" in the development of +, &c.). It is enough to satisfy this condition for each letter nnd its preceding one ; that is to say, each coefficient differentiated with respect to any one letter, is to yield the same result as the directly preceding co efficient differentiated with respect to the directly preceding letter.
The following rules are found sufficient. To pass from any one de rivative of pa to the next, arrange the letters a, 6, e, &c., or a„ a., &c., whichever may be used, in order, in every term : differentiate with respect to the last letter in each term, and multiply by the letter which comes next to IL And when the last but one immediately 'precedes the last in the alphabet or other consecutive system, do the same with the last but one, and divide by the exponent of the last letter, as it becomes after the increase which it receives from the process of the preceding letter; but in no case use any letters but the last or the last but one. For instance, beginning with Oa, in which is only one letter, we have . b, or nsba=tga.b; in which are two letters, a and b, consecutive. Operate upon b, and we have is'a e; operate on op'a, and we have again ca'a .6, which, with the b which was in before, is '"a. i.e, which we divide by the new exponent of 6, or by 2, whence 0"a /Apex = 0'a + •=, In forming trlpa, we use only e in Va. e, because a does not imme diately precede c; and we get (the succession being a, b, e, e, f, g, &c.) Dada = + . 214 + P; and so on. As soon however as the law is established, it is best to form a table of the successive derivatives of the powers of 6 by this same ltw : we then have D" oct = Van"?' + 2 2 . 3 — D"_ -G' + — +dc.
•a as far as Go • ) in which op'a, fra, &c. are to be taken from the function by common differentiation, and the derivatives of the powers of G from the table. This being done, we have fga + + + Ix' + gxs +, &c.)= + and the process is shortened to its utmost extent ; all that is not differentiation being merely reference to a table and writing the result.