OPERATION. 



OFE1UT10N. 



A ^-r"*-r- of the preceding notation may be made as follows : 

 if K or ' he simply adsreataon to ioorsase * by unity.and * a direction 

 to let H remain unaltered, it is dear that n etf most mean (x+ 1 + 1 1, 

 or t, *r ,- so that * \r and e, ** are tae same. Similar reasoning 

 SHISJIII to it, whuiTsf * is a whole snlin ; and shows that it is 

 c and nothing els*. SinOar reasonins; also applies to - where A is 

 a whole number: for r-* must be so interpreted that r performed 



upon K may give *-* or Jf; that U, e- f* with r changed h times 

 into * + 1 must be r +r, at xr ; or - *r must be (z - A V In 

 like manner it may easily be shown that our of the mfaxtngt of E*.^r, 

 where A fa fractional, is f (x + *) : bat, as in common algebra, of 

 which all the conclusions, a* shown, here apply, when A in a fraction, i ' 

 may be any one out of operation* a* many in number as there are 

 unite in the denominator of A. To take a very aimple case, required 



K t (r), meaning an operation which, twice repeated, gires B* Qx, or 

 4 (x + t). Thi* condition i evidently satisfied by <t> (x + Ji), but it 

 u ibo smtisned by j(x + 4*"V f r >' P art f l ' u ' operation consist in 

 ) of aign, two repetitions of the operation reproduce the original 



It rnuat not be forgotten that, in finding new objects of algebraical 

 reasoning, we hare not lost our right* over the old one* ; thus any 

 latter may stand for a multiplier or divisor of the universal subject of 

 calculation, fcr, But thoe ideptr>de*t mulHplieri mutt nut contain .T : 

 for if they did contain x, the convertibility of the multipliers with E 

 would not any longer be practicable. If, for instance, we consider 

 a Kfur, which ia a $ (x + 1 ), we find it to be the same as E (ajfu-), for a, 

 being independent of r, is not affected by E. But if we consider ft.<t>r, 

 and E (.rfa:), we find the first to be xf (.r + 1 ), and the second to be 

 (i + \)f(j- + l\. A wide branch of the calculus of operations exists, in 

 which the convertil-K- character of the operations is not made a postu- 

 late. Our limits will not allow us to enter upon it, except to a very 

 amall extent, as presently given : )>ut there is one elementary work on 

 the subject, Carmichael, 'Calculus of Operations,' 1855 ; and there is a 

 good deal on the subject in Boole, ' On the Calculus of Finite 

 Difference.,' 1860. 



To generalise the preceding, we may suppose Er to mean Qx as 

 before, and x^x to stand for <f> (x + a), where a may have any value we 

 please independent of jr. And it is in our power to abbreviate any 

 collection of operation* by using a single symbol to stand for it. Thus 

 v^x to stand for such a set of operations as 



which we should denote by 



V=A -rA,E+A,E l + ---- 



Again, if in common algebra fy were AO + A.JT + ....,we might 

 abbreviate the preceding into /E instead of y. 



But it may be *aid that thi, though intelligible as to a simple 

 operation E, its repetition* tf, E 1 , &c., ita inverse and repetitions of it 

 K-I, -*, Ac., eaaae* to have meaning when we come to apply it to 

 other function* of algebra. What, for instance, is log. (1+1)1 How 

 can the direction to add f(x + a)to$x have a logarithm ? This ques- 

 tion arise* from the student having carried with him into purely 

 symbolical algebra (in which it is the first requisite to drop all 

 ing*) signification* of symbols derived from ordinary arithmetical 

 algebra. Now it ia to be remembered that as far as we have yet gone, 

 all the transcendental symbol* of algebra, a-*, log E, sin E, cos E, &c., 

 hare not been mentioned, far less denned; they are not therefore 

 absurd, bat only, for the present, unmeaning. The question is, how 

 are we to give them meaning ; at our pleasure, or by deduction ? 

 Evidently the Utter, for we are bound to retain the power of n-ini; 

 algebraical transformations as they now exist. Since then a* in 

 common algebra i* equivalent to 1 + logo.jt+4 (log a.)*x 1 + ...... we 



must lay it down that a = 1 + log a. E + . . . . , or that a * must be 



viewed, when it means an operation to be performed upon <px, as an 

 abridgment of 



fix + log a. x+z + J (logo.) 1 E : ^r + ---- 



This is unquestionably the most difficult step of the whole : we shall 

 have occasion further to consider it in the article RELATION, but for 

 the present the following may be sufficient. Since the total operation 

 AO + A,E + A,E* + . . . . can be easily understood, consisting merely of 

 the successive performance of the operation E, the multiplication of 

 the result* by given quantities of common algebra, and the addition of 

 the products ; and since all the transcendental* 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 transcendentals they 

 stand fur ; let us consider the transcendental symbols of operation as 

 abbreviations of the series, supposed to stand for series of operat 



We rhaH now proceed to some examples. First, let it be required 

 to transform the series A^SUC + A, *(*+ a) + A,f (.r + 2a) + ---- This 

 may be represented by A,,-*- A. E + A,E' + . . . performed upon 4*. Let 

 the latter series in common algebra be ft, then /K, considered as a 

 symbol of operation, stands for the preceding complex operation. 

 Let the transformation be required to be made into a series of terms 

 containing fa- and Ha differences : let f (x + o) <pe = A^ac, then E 1 



is A, or i = l > A. But/Eor/(l+A) is B O + B,A + B,A ! + ---- where 

 n n . B, , Ac. are the values of .r and its successive differential coefficients 

 when y = 0, divided by 1, 1, 1 . 2, 1 . 2 . 3, Ac, Consequently the pre- 

 ceding series is the same as n,,$.r + B,A^r + E^A'fx + ---- For instance, 

 let the series be j>x-+(x + a) + <t> ( j- + 2o) . . . or (1 K + E' ---- ) ^.c 

 or (1 + K)- 1 <ff. Write 1 + A for , and we have (2 + A)- 1 +x or 

 .111 



We have chosen this result as one which i* very easy to verify. Let 

 <tu-, $(.<: + a), &,c, bo denoted by X , x,, &c., then [Din IIHF.NCKS] we 

 have 



svr=x , A^r=x 1 -x a . AV = x.-2x 1 +!, 



A*4w = x, - 8x . + 8x , - s , Ac. 



Substitute these in-jj $z -7 &<pjr + . . . , and we have 7; -. 



(x, x<>) + . . . . : take a few terms, developing the fractions multipli- 

 cation, and we shall find the preceding to be identical \vith 



111 \ /I 2 3 



6 

 32 



/I 3 6 

 U + ll + 32 



2 V '21 X " 4 V 1 ~" 2/ X ' + 8 V 2/ X ' ~ 



or s,, - x, + x, - . . . or q>x <f> (x + a) + <f> (x + 2a) - . . . 



We shall now take an example of interpretation. Requirnl tin- 

 meaning of A-> by means of El Since A=E 1, we have A-' = (K ]) 

 -i E-' +E-' +E-* +. .. . or 



A' $>.r means <j> (x o) + <f> (x2a) + . . . . ad ifi< 



This is easily shown to be consistent with the relation AA~ ' = 1 or 

 AA ' <t>x= </>.r, for if the preceding series be called >(/.<, \vo have A A - '$w 

 = A<)>x=il>(x + a) ^x^(^a; + <t,(j'-,,)-t- .... )-(( )+$(*) 5U) !- 

 . . . . ) = ^r. We have thus obtained and verified one of tin- infinite 

 number of forms which A' <f>.r may represent. 



As yet we have nothing 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. 



Let (<f>(.r + 9) iftx) : t be called D. : then the smaller 6 becoin. 



more nearly ia D.^-r the differential coefficient of <}u?, or <p!x. Let 6 be 



the th part of the given quantity a : then the smaller 9 is, the greater 

 must n be. We have then 



; since 



= (l + D a 



Proceeding in this way we obtain $ (r + H ff) or 41 (x + a) = (I + 9v ) A.C 



n--l 

 = i>x + nBD.&x + n S'D' AT t . . . . For n9 write a, and the 



V V 



preceding becomes 



which being always = (x+ a), has a limit also = (r + a). Take that 

 limit by diminishing 9 without limit, and we sec that n.<)>.r, D^.r, &c., 



become <t>'x, <p"jc, &c., or 



which is Taylor's Theorem. Suppose we denote the operation of 

 differentiation by D, and ^ (x + a) <f>x by A^.T, we have then 



aparticuhvr case of which (when o=l) was chosen as onr illustration 

 at the beginning of this article. This relation 1 + A = t"* gives us a 

 great power of converting series which contain differences into those 

 containing differentials, and rife rtrsd. 



\Vi- now propose to interpret D-'. This symbol must satisfy DD~' 

 q>x=$j-, and D~' D<px=q>r, the first of which is satisfied by o~ > ^ix = 

 Jfxdx + c, where c may have any constant value: but the second is 

 only satisfied byf<j>xdx, beginning at a value of x which maks <j>;r = 0. 

 We shall however see that we need not enter on this question in 

 reference to the theorem immediately following. 



Let it bo required to express A~'<)>2:, or $(x a) + <t>(x 2o) + . . . 

 orf inf., by means of operations of the differential and integral calculus. 



