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number of terms, the only difference in the absolute equality of the two sides of the equation is that which arises from 0 being an undetermined fraction, mean between 0 and 1. The last term of (81) is called the limit of Taylor's Series.

73.] Examples of Taylor's Series. Ex. 1. f(x) = log, a?.

fix) = i = x~\

/"(*) = (-)tf-2,

rw = (-)*i.2*-s,

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Section 4.—Change of the equicrescent variable, and transformation of differential expressions.

75.] Prom the supposition which we are at liberty to make that one of the variables involved in an equation should increase by equal increments, and therefore that the several differentials of it, after the first, should vanish, problems such as the following arise:

(1) Suppose y = f(x), and that an expression is given involving x, y, and some of the derived-functions or differential coefficients which have been calculated on the supposition that one of the variables is equicrescent; to change the equation into its equivalent, when neither of the variables is equicrescent. Or

(2) To transform it into its equivalent, when the other variable is equicrescent. Or

(3) An expression being given involving a variable, which is either equicrescent or not, and its differentials, and also an equation being given connecting this variable with some other new variable; to eliminate the old variable and its differentials by means of these two equations, and to replace them in the original equation by their equivalents in terms of the new vari

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able: the new variable being equicrescent or not, as the case may be. Or

(4) It may be required to replace the variables and their differentials in a given differential expression, by their equivalents in terms of new variables, which are connected with the old variables by means of a sufficient number of given equations: the old and the new variables being equicrescent or not, as the case may be.

All these several processes involve transformations of differential expressions, and because such expressions commonly involve second and higher differentials, and one of the variables has been assumed to be equicrescent, they are called changes of the equicrescent variable. The method of effecting them is the same in all cases, viz.

To replace the expression, which has been simplified by the condition of a variable being equicrescent, by its complete value when no such modification has been made, and then to introduce the other conditions which the problem requires.

76.] Thus to solve the first two of the four cases above.

Let the given expression involve x, y, ^, , ;the

differential coefficients, as their form indicates, having been calculated on the supposition that x is equicrescent; it is required to replace these several differential coefficients by their equivalents, when x is not equicrescent. Since by equations (2) and (3), Art. 54,

therefore if x is not equicrescent,

dx2 dx dx9'

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77.] In (3) and (4) of the cases of Art. 75, suppose the given expression to involve y, x, wherein x is the equicrescent variable, and suppose the new equation to be of the

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