Let (cp(x + 0)— cpx): 0 be called : then the smaller 0 becomes, the more nearly is Doty the differential coefficient of or, or itix. Let 0 be the nth part of the given quantity a: then the smaller 0 is, the greater must n be. We have then + O.r, (x+ 2 0)= (1 + since 4)(x + 20)-0 (x+ or (r+ 20) = (1+ (.e+ = (1 ib.r.
Proceeding in this way we obtain cp (x or cp (x+ a)= (1 + OD 4 ox n-1 = ox + + n + For n0 writo a, and the preceding becomes Ox a +. .
which being always =0(x+ a), has a limit aleo= (x + a). Take that limit by diminishing 0 without limit, and we see that &c., become o'x, ca".r, &c., or (x+a)=cpx+ib'x.ci+ib".r +..
which is Taylor's Theorem. Suppose wo denote the operation of differentiation by n, and cp (x + a) - elix by apx, we have then i+Li=i+XD+=Esp.; 2 °particular case of which (when a=1) was chosen as our illustration at the beginning of this article. Thin relation 1+ A= gives us a great power of converting series which contain differences into those containing differentials, and rice rend.
We now propose to interpret Thin symbol must satisfy tPX=OX, and the first of which is satisfied by + c, where c may have any constant value: but the second is only satisfied byfoxdx, beginning at a value of x which maks Ox = 0.
We shall however see that wo need not enter on this question in reference to the theorem immediately following.
Let it be required to express it-'˘x, or 0(x-a)+O(x-2a) + . . . ad inf., by means of operations of the differential and integral calculus.
Since a is e"-1, we have to find expanded in powers* of D. Now common algebraical processes show that (a t —1)-1 can be developed in the series 1 1 t to is - + 131 75. - + 23 2.3.4.5.6 where Bi, &C., are the NUMBERS OF BERNOULLI, of which an 1 1 ample stock is given in the article cited : thus = , etc.
Write an for t, restoring cpx, and for nos, die., write cp'.r, &c. but for writefserdx + c. We have then (x-e) + ˘ (x-2a)+ (x-Sa)+ . ad Irvin.
1i• 1 = a- 5. + 2 +.
The determination of the constant might in many eases be trouble some, but if we only want a finite number of terms of the series, we can avoid it altogether as follows. Let .c-na=y, and suppose that 4)(x -a) + . . . . ending with cp (x-na) is sought. Write y instead of x in the preceding, remembering that y - a = x- (a + 1)a, : tract the result thus obtained from the preceding, and we have (x -a) + (x -2a) + . . . . + cp (x-na)
1 1 B,a = J .rdx - +7 ($'s • • • • Butigardr -jcp.rydyisfordx taken from y to x, or if cp,x differen tiated give ox, it is cp,y. We introduce this process merely to give some idea of the process to the reader who is already master of the result from other sources; and we cannot here explain the reason why one particular form of cpx is taken.
For further developments of the results of this subject, see the Appendix to the Translation of Lacroix ; lierschera 'Examples of the Calculus of Differences ;' Lacreix'e largo work on the Differential Calculus, voL iii.; Library of Useful Knowledge-Differential Calcu lus; a paper by Mr. Murphy in the Phil. Trans. for 1837; and various papers in the numbers of the ' Cambridge Mathematical Journal.' In several of these works further references will be found. Many other references might be given to scattered volumes: but the student will find much collected together in Professor Boole's two recent works on Differential Equations and on Finite Differences. The student may make an attempt at the demonstration of the following theorem a teat of his understanding the method which we have explained, and the points of analysis which are most essential as preliminaries. Whatever a may be, the differential coefficient of ipx is an algebraical I equivalent of the following series: cp (x + a) -cp (x -a) 4: (x +2a) - cp (x - 2a) a 2a (x + 3a) - cp (x - 3a) 3a Instead of supposing Or, a function of s only, to bo the fundamental subject of operations, we might make it cp (x, x,) and suppose E and E, to represent the operations of changing x into x + a and r, into .r, + a,. We can only briefly note some of the results of this extension. If D and D, represent the operations of differentiation with respect to x and to x„ we have in the means of obtaining the common extension of Taylor's Theorem to a function of two variables. Again, if we take 0.1-spx, and let D and D, represent operations of differentiation, which separately affect cpx and we have in the development of (n+ n,)",p.r.f.r, the formula for forming the nth differential coefficient of the product.