(a − b) (a' — b′) (a′′ – b′′) = a a' a" cos 0 cos 0' cos2 0′′. sions of 897. There are many applications of analysis, particularly in ConverAstronomy, in which it will be found to be extremely useful to other forconvert expressions, such as a sin Ab cos A, a cos Ab sin A, mulæ by means of subsidiary angles. where is less than a sin3 A, becomes, by making b = a sin3 0, a modification of its form which is not unfrequently used. CHAPTER XXXV. ON EXPONENTIAL AND LOGARITHMIC SERIES. Develope- 898. THE exponential expression a, the series into which ment of a. it may be developed, and the various logarithmic and other series which may be deduced from it, enter very extensively into analytical enquiries, and deserve the most careful examination of the student. We shall begin with the developement of a* into an equivalent series. If, in a, we replace a by 1+ (a−1), making a = {1 + (a− 1)}*, we shall find, by the binomial theorem, (Art. 680) if we actually multiply the factors of the exponential coefficients (Art. 688), and collect together the terms which severally involve the same powers of x, denoting their successive coefficients by k, A2, A3, A4, &c., we shall get a series which possesses, as we shall afterwards shew (Art. 906), some important properties; it remains to determine the coefficients A, A, ... which, when severally divided by the terminal (Art. 688) coefficients 1 x 2, If in equation (1), we replace a by y, we get If we multiply together the two sides of the equations (1) But if, in equation (1), we replace x by x+y, we find which becomes, by expanding the successive powers of a+ y which it involves, q2+9=1+kx+ A1⁄2‚22 ̧+ As1.2.3 x3 + ... 1.2 We thus obtain two forms of the developement of a**", which can only become identical in form as well as in value, when the respective terms of both series which involve x and y similarly are identical with each: thus if we compare together the terms in the two series which involve y only, without its powers, we find of identity lent series. if we replace A2, A3, A4,... by their values, we get a series in which k and x are similarly involved. Conditions 899. The principle which is involved in this investigation of equiva- is one of great importance, and capable of very extensive application: it is presumed that the form of the series which is equivalent to a is independent of the specific value of x, and therefore the same when x is replaced by y, or by + Y, or by any symbol or combination of symbols whatsoever: but if a*+ be equivalent to the product of a and a for all values of x and y, then likewise the series for a*+* must be equivalent to the product of the series for a and a" under the same circumstances: and this equivalence of the results which are obtained implies that they are identical in all those terms in which x and y, one or both, are similarly involved: we are thus enabled to obtain a series of equations, expressing the conditions of identity, by which the form, or analytical values of the successive coefficients are determined. Is the existence of for a ne cessary or not? 900. It may be further observed that the existence of an the series equivalent series for a*, or of a series which shall possess the same analytical properties with a", is a necessary consequence of the binomial series in its general form, and involves "the principle of the permanence of equivalent forms" no further than it is involved in the binomial theorem: but when the binomial or any other theorem is once established, whatever be the principle upon which it rests, it becomes one of the known and acknowledged results of Symbolical Algebra, and may be employed in the deduction or establishment of other conclusions equally with the results of the definitions of Arithmetical Algebra: it is thus that the bases of Symbolical Algebra are perpetually enlarged, and the great principles which present themselves in the first and most elementary of the generalizations which it requires, are speedily replaced by other and successive links in the long chain of consequences which are found in the progress of our enquiries: such results, therefore, though their existence per se may not be necessary, yet become necessary results when considered with reference to each other and to the propositions upon which they are finally dependent. a rapidly converging series, from which its numerical value may be calculated to any required degree of accuracy: the aggregation of 14 of its terms gives us which is correct as far as the last figure: but no finite decimal number can express its accurate value*. It is usual, in all cases, to denote this number 2.7182818 by the symbol e; it is the base of Napierian logarithms (Art. 855), which are exclusively used in analytical formulæ. The process for this purpose is very simple and expeditious: divide 1, and the successive quotients which thence arise by the successive natural numbers: the sum of the quotients, increased by 2, is the number required: thus, 1. 2) 1. 3) .5 .16666666 4) .04166666 5) .00833333 6) .00138888 7) .00019841 8) .00002480 9) .00000276 10) .00000027 12) .00000000 2.71828180 It has been shewn, in Art. 203, that e is an incommensurable number: it is not capable, therefore, of being expressed by any finite decimal. The series for e*. |