To genemliee the preceding, we may suppose EN= to mean ox as before, and Box to stand for ¢ (x+ a), where a may have any value we please independent of r. And it is in our power to abbreviate any collection of operations by using a single symbol to stand for it. Thus Laplace uses Vier to stand for such a act of operations as A„ite(x)+ s.,+(x+ a) + sotqx +2a) + which we should denote by V • = A.+ a, E + A5 , +Again, ii in common algebra fy were cs„,+ a, y + ...., we might abbreviate the preceding into fE instead of v.
But it may be said that this, though intelligible as to a simple operation E, its repetitions &c., its inverse and repetitions of it kc., ceases to have meaning when we come to apply it to other functions of algebra. What, for instance, is log. (1 +E) I How can the direction to add ip (x + a) to ox have a logarithm / This ques tion arises from the student having carried with him into purely symbolical algebrh (in which it is the first requisite to drop all mean hap) uignifications of symbols derived from ordinary arithmetical algebra. Now it is to be remembered that an far as we have yet gone, all the transcendental symbols of algebra, ece, log E, sin E, cos E, &c., have not been mentioned, far leas defined ; they are not therefore absurd, but only, for the present, unmeaning. The question is, how are we to give them meaning ; at our pleasure, or by deduction / Evidently the latter, for we are bound to retain the power of using algebraical transformations as they now exist. Since then a' in common algebra is equivalent to 1+ log a. x + b (log we must lay it down that a' +log a. E +...., or that a' must be viewed, when it means an operation to be performed upon ,pr, as an abridgment of Ox+ log a. nrPz+ I (log +....
This is unquestionably the most difficult step of the whole : wo shall have occasion further to consider it In the article RELATION, but for the present the following may be sufficient. Since the total operation can be easily understood, consisting merely of the successive performance of the operation a, the multiplication of the results by given quantities of common algebra, and the addition of the products; and since all the transcendentah of common algebra can be expanded, in series of the above form, in such manner that the series have all the algebraical properties of the transcendental. they
stand for; let us consider the transcendental symbols of operation as abbreviations of the series, supposed to stand for series of operations.
We shall now proceed to some examples. First, let it be required to transform the Penes a.rox + •, 0(x+ a) + (x + 2a) + This may be represented by E + + ... performed upon Or. Let the latter in common algebra be fa, then fa, considered as a symbol of operation, atands for the preceding complex operation. Let the transformation be required to be made into a 'series of terms containing ox and its differences: let cp (a- + a) - Or= a cpx, then E-1 is a, or a=.1 + e. But f E or f (1 + a) is n,A + . . where n„. n„ &c. are the values of fy and its successive differential coefficients when divided by 1, 1, 1 . 2, 1 . 2 . 3, ke. Consequently the pre ceding aeries is the same as a„cpx + + no'cpx + .... For instance, let the series be +.e-+ (x + a) + cp(x +2a) ... or (1-a + a'-....) or (1 + or. Write 1+ A for E, and we have (2 + Or or I 1 I (7, - + g - Ix, or - + Witty We have chosen this result as one which is very easy to verify. Let or, (x+ a), cte., bo denoted by x., x„ &c., then [DierEnucr.s] we have Lox =x,—;„ = x,-2x,+x„, Asox= x.-3x, +3x, -x,„ ke.
1 1 1 Substitute these in7, Or - Ltep.r. + , and we have 7, o (x,-x,,) + take a few terms, developing the fractiona cation, and we shall find the preceding to be identical with (1 1 1) ( 1 2 (1 3 61 ( 1)-i 1 ( 1)-: 1 ( 1)--.
— — x, + - x, or ; - x, + x, - .. or ox - + a) + (x + 2a) We shall now take an example of interpretation. Required the meaning of by means of Er Since A=E-1, we have = I + +.... or a-' cps means o (x-a) + o (x -2a) + ad infinitum.
This is easily shown to be consistent with the relation =1 or AA-I ox= or, for if the preceding series be called Or, we have .pf.r=4. + a)-4a= (cps cp(r-a) + . . . . )- (0(x -a) + o(x - 2a ) + )=or. We have thus obtained and verified one of the infinite number of forms which A-I or may represent.
As yet we have nothiug which might not have been done with tolerable ease by common methods, nor shall we have done more in proving Taylor's Theorem, but the step which we shall make to follow that proof will be an instance of the deduction of a theorem which is of a more difficult character.