4'J 



TAYLOR'S THEOREM. 



TAYLOR'S THEOREM. 



to 



the first dimension, every terra of D" If is of the rth dimension ; but if 

 we consider each letter as of the dimension following : 

 bcefghklm n p q 

 12345678 9 10 11 12: 



then every term of D*b r is of the ()i + r)th dimension. To find out 



if all the proper terms be there, and with the proper exponents, write 



down the number of ways in which n + r can be made out of r num- 



bers. Thus, to verify this point for D 7 4 3 , write down the ways in 



which 10 can be made out of three numbers, namely : 



8 + 1 + 1, 7 + 2 + 1, 6 + 3 + 1, 6 + 2 + 2, 5 + 4 + 1, 5 + 3 + 2, -4 + 4 + 2, 



4 + o + 3 ; 



take the letter answering to each number, in the above list, and 

 multiply the letters of each set together, which gives 

 If I, IA, Ith, <fk, bfy, ceij, r/-, e-f, 



which are, coefficients excepted, the terms of D'4- 1 in the table. To 

 verify the coefficients separately, observe that the coefficient of that 

 term of D* It' which contains the Hh power, (th power, &c., is 

 1 . '1 . 3 . .._. (r - l)r _ ' 

 1.2.3...*xl.2.3...tx ..... ' 

 Thus, in D*4 8 , the term containing 4Ve ought to be multiplied by 



1.2.3.4.5.6.7.8 



; r - T -. :, or 168, as is the case. 

 1.2.3.4.0 x 1.2x1' 



But the best general mode of verification is derived from the 

 theorem 



Ido'b' 1 fdD* I' \ 



&-'- - P -3T, "' 6-' ' ; D l^4-J ; 



that is, having a certain derivative of a certain power, the next higher 

 derivative of the next lower power may be found by differentiating 

 with respect to 6, dividing by the exponent of the original power, and 

 then performing the derivation. Thus : 



D"6 = 94'/+ 724'ce + 844V, 

 differentiate with respect to 4, and divide by 9, which gives 



Now derive, which gives 



Sb'g + 564V/+ 284V + 1684Ve + 704'c', 

 the same as is found in the table for D'+'fc 5 -'. Here we verify the 

 i iuli.-r result of the table from the later : to verify the later from the 

 earlier, use the following : 



D" I' = D*-' e.rlf-* * D"- J c 2 , r , b*-* + , &c. . 



up to e" 



., 



' ar- 



n derivation. Let the reader try for himself (4 + CZ + CT-+/X 3 ) 5 by 



bis mode and then in the common way, going only so far in the latter 



as to feel sure that the former is of no trouble compared with it. Let 



i, m " ! ~ , &c., be denoted by m, m v &o. 



(a + bx + ex- + &c.)' = (i 1 * +m4a nt ~ 1 .i; 

 + (mac + iiiji") a m ~-x'* 



+ m^aob 3 + m t 4*) a"" 4 x 4 +, &c. ; 

 he law of which is evident, the only thing left being the substitution 

 the values in the tables instead of the derivatives of It. This form 

 s convenient for fractional or negative powers. The following case is 

 worth exhibiting separately : 



1 14 l-ac a 



7 = ~ ~"5 x + - x " 



, &c. 



. 4-e 



a a- ~ a- 



4 a D4 3 + a'D**" 3 / 



z*-,&c. 



a a 5 



We have avoided the formality of writing D4 for c, D C 4 for c, &c. 

 A + B.c + ex- + , &c. A A4-BJ 



:c. a 2 



'-ac) 



A (4 - ao4 + a'<) - BO (4' ni) + ca'4-.Eo 3 



- x -\- . etc 



o 



The law is here evident enough ; the next numerator would be 



The derivatives of the general term 4* may be readily formed, but 

 the particular cases are more useful ; see the derivatives of a in the 

 general form above given. We shall not overload this subject with 

 further examples : enough have been given to show those who require 

 developments of some extent how much labour they might save. 



We shall conclude this article by recommending that the process of 

 derivation should be introduced, without demonstration, of course, 

 into elementary books of algebra, as one of the best exercises of simple 



in which the derivative* of powers of e must be formed from th< 

 'ilar ones of 4, by changing each letter into the nex 



ing. There are thus abundant means of verification. We wil 

 mention yet one method more. Only the last letter and the last bu 

 ana (and that only when the two letters are consecutive) are used in 



rivations. If we use any letter, no new term is produced, bu 

 only a repetition of those which other terms give. For instance, i 

 D4 5 is the tenn 604*a/; and in passing to D'4 5 , we derive from 

 because it is the List letter ; and from e because, being the last bu 

 one, it immediately precedes / in the series. We do not here use 

 and c at all ; but if we did use them, we should only repeat terms 

 which will come into D'4 1 from other sources. Thus : Wlfcef gives, 

 from /, 60lfce;i, which U set down in D T 4 S ; from e, 604V//-:- 2, or 

 304 : c/-, which is also set down ; from c, if c had been used, we should 

 have had 604*/-=-2, or 304V/, which, on looking, we find set down, as 



j, from the last letter of 104V. From 4, in 604V/, had it been 

 used, we should have got 1204rc</4-2, or 604c = </, which is also found, 

 and arises from the last letter of 306cV. If then we ever find tliat 

 derivation from one of the unused letters gives anything but what 

 arises from some of the letters which are used, it is a sign that some 

 error has been committed. 



By help of the preceding method, expansions which analysts usually 

 avoid as much as possible, at almost any expense of circuinoperation, 

 are carried on with the greatest facility even further than is necessary. 

 The development of <t>(a + b.r + cx 3 + &c.),alrcady given, is one instance; 

 the process in REVERSION* OF SERIES is another. This last is done by 

 expanding x in powers of ax + bx' + , &c., by Burmann's Theorem, and 

 making the expansion of the negative powers of (a + 4x + cx i + , &c.), 

 which will be wanted, by the method of derivations. We shall state 

 some further applications : 



When m is integer, these derivatives are in the table. 

 6 + r.+,&c., is n finite series, the whole result is brought out with 

 great ease, compared with the trouble of the common algebraical 

 operation : in this case, the value of every letter after the last in the 

 finite series is 0, or the lat letter of that series is not to be employed 



ARTS AND SCI. DIV. VOL. VIII. 



one another in rapid succession with much sameness and some diversity. 

 For this reason we should recommend, in arithmetic, Horner's process 

 [INVOLUTION AND EVOLUTION] ; and in algebra, Arbogast's derivation. 

 We proceed accordingly to divest this method of the phraseology of 

 the differential calculus, and to put it before the elementary student in 

 algebra. 



The name of the process is derivation ; its primary object the 

 raising of any power of an expression of the form 6 + ex + c.ir +fx 3 + , 

 &.C., immediately that is to say, by writing down the result at once, 

 without any but simple mental processes in passing from term to term. 

 The rules are as follows : 



1. Begin with that power of 4 which is to be raised. 



2. To pass from the coefficient of one power of x to that of the next, 

 multiply each letter by its exponent; then diminish that exponent by 

 a unit ; then introduce the next letter. And if this last process 

 increase an exponent, owing to the letter newly introduced having 

 been in the term before, divide by the increased exponent. Hut 

 remember never to operate on any letter except the last in the term, or tlie 

 latt but one ; upon the last always, upon the last but one when it 

 immediately precedes the last in the original series 4, c, e,f, &c. 



3. If 4 + c-r + ,&c., be not an infinite series, but a finite number of 

 terms, operate as if the succeeding letters were severally equal to : 

 for instance, if y be the last letter, drop every term in which h should 

 appear, as fast as it arises. 



For example, the fifth power of b + cx + cj?-l-f.c: Begin with 4=, 

 derive from it 54*r, the two first terms are 4 s + 54'c . x. 



To form the coefficient of j?, take 54 4 c, and observe that 4 and i: 

 follow each other in the series, BO that in the next derivation there are 

 two processes. First, use c or c 1 , the last letter, which by the rule 

 gives lce or e : so that derivation applied to the first power of a letter 

 gives merely a change of that letter into the next : hence 54 4 c gives 

 5ft 4 *. But 4 4 , which must also be used, gives We, and 54 4 c gives 

 5(44'\-)<;; so that c becomes c-, and we must therefore divide by the 

 increased exponent 2, giving 104 3 c'-. Hence the next term is 

 (54'e + 104-V).r-. 



In the next derivation 54 4 gives only 54 4 /, for 6 not immediately 

 preceding in the series 4, c, c, &c., is not used. But 104V gives 



p 2 

 , or 204'cj + 1 04V. 



1 04'(2) + 



Next term (54'/+ 204 3 *: + WbV).>-\ 



In the next derivation 54*/ must be neglected entirely, because / is 

 the last letter, and 4 is not the one immediately preceding. Also 

 204 3 cc gives 204'e/ and 204 3 e-^2, or 104V; while 104V gives 804V 



