45 



OPERATION. 



OPERATION. 



46 



Since A is !, we have to find (t ta l)~ l expanded in powers* of 

 D. Now common algebraical processes show that (c* 1)~' can be 

 developed in the series 



lit* 3 t> 



" 5 2. 3. 4. 5. 6 



-&c., 



where BJ, B 3 , B 3 , &c., are the NUMBERS OF BERNOULLI, of which an 

 ample stock is given in the article cited : thus B, = -g , B, = g^, &c. 



Write OD for (, restoring <t>x, and for Dfx, &c., write <t>'j;, &c. but for 

 D~ ! tja: vnitefipxdx + c. We have then 



tj> (x-a) + <t> (x 2a) + <*> (x 3a)+ ... ad 



1 B.a B, a 3 



O- g ** **-2737i*"'* + - 



The determination of the constant might in many cases be trouble- 

 some, but if we only want a finite number of terms of the series, we 

 can avoid it altogether as follows. Let x na=y, and suppose that 

 $(x a) + . . . . ending with $ (xna) ia sought. Write y instead of 

 x in the preceding, remembering that y-a=x (n+ l)o, &c. : sub- 

 tract the result thus obtained from the preceding, and we have 



$(jt-o) + (js-2a 



;/**- j/fi*r-5< 



<f> (xna) 

 (t'x-t'y) 



: taken from y to x, or if ^x differen- 

 tiated give 0J-, it is $,* ^,y. We introduce this process merely to 

 give some idea of the process to th reader who is already master of 

 the result from other sources ; and we cannot here explain the reason 

 why one particular form of A~' <t>x is taken. 



further developments of the results of this subject, see the 

 Appendix to the Translation of Lacroix ; Herschel's ' Examples of the 

 Calculus of Differences ; ' Lacroix's large work on the Differential 

 Calculus, voL ili. ; ' Library of Useful Knowledge Differential Calcu- 

 lus ;' a paper by Mr. Murphy in the Phfl. 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 

 inch 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 prclimiiun.--. 

 Whatever a may be, the differential coefficient of ^ is an algebraical 

 equivalent of the following series : 



2u 

 -(j; 3a) 



Instead of supposing $s, a function of .r only, to be the fundamental 

 subject of operations, we might make it $ (x, x,) and suppose E and i:, 

 sent the operations of changing x into x + a and .r, into r, + , . 

 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 f lt we have in 



EE, = fH ..= ! + (aD + a,D,)+ .... 



the means of obtaining the common extension of Taylor's Theorem to 

 a function of two variables. Again, if we take Q.'-fyx, and let D and D, 

 represent operations of differentiation, which separately affect f.r and 

 ijw, we have in the development of (D + V,)*QX$X, the formula for 

 forming the nth differential coefficient of the product. 



We shall conclude by some indication of the principal step* of the 

 application of the calculus of operations to the solution of differential 

 equations and equations of differences. 



However 'much the calculus of operation* may throw light upon 

 the character and principles of algebra, it would at one time have 

 been thought unlikely that it should much facilitate actual processes. 

 It does this, nevertheless, and nowhere more than in the subject we 

 are now going to describe. Solutions which by the usual method 

 VTION or PARAMETERS] would never have been considered fit 

 examples for an elementary work, on account of their complexity, 

 may be represented with ease, and obtained in full with very little 

 trouble. 



A\ lien rules of algebra are true of the meanings of any symbols, all 



consequences of the use of these rules, all relations which are legiti- 



!>'iliu;t,i"iM from tin-in, til 'i represent truth*. Not that these 



truth* are always intelligible without si: interpretation: nor 



mean to say th-it, in the present state of the science, the inter- 



:nns are always attainable. And further, it may happen that 



nu can bo pointed out, derived from processes in which tome 



* We trtnifcr tUia word, with extension of meaning, to the calculus of 

 operation*. 



only, and not all, of the fundamental rules of algebra are true. This 

 does not prevent our right to deduce conclusions from such theorems, 

 so long as we use no fundamental rules except those which are true of 

 the expressions in question. For instance, we have seen that the 

 operations of our calculus are not convertible with the operation of 

 multiplying by a function of the variable. Thus if E stand for the direc- 

 tion to change x into x + 1, A for that of forming the difference thence 

 arising, and D for the direction to take the differential coefficient with 

 respect to x, we have no right to say E</U:(I|/X) = QxEtyx), or A(px(<f/x) = 

 tyx&tyx), or Dipx (tyx) = tyxT>($x) ; in which tyx is the function operated 

 upon. But, when we thus use another function, </>.r, besides the one 

 operated on, tyx, this convertibility of operations is the only rule of 

 algebra which fails ; it is therefore the only one the use of which we 

 must avoid. 



The operations E, A, and D, are closely connected with E+a, A + a, 

 D + a, of which they are particular cases ; a being a constant, positive, 

 negative, or nothing. We have 



(E a) . <fix = 0*+' A (a-* <fu) 



(A - a) <t>x = (a + 

 (D a) $x = t" D 



((a -I- l)-*<?.r) 



The first sides of these equations being representatives of <f>(x + l) 

 a$x, &<l>xa<t>x, and <j>'x a<j>x. If these operations be repeated, we 

 have 



(E - a)"<t>.c = a'+A" (a-*(fa:) 



(A - af^c = (a + l)*+"A"((a + l)-*^-) 

 (D a)* (fix = e"D* (c-" <px). 



These results will also be found to be true when m is negative, by 

 which means we are enabled to interpret (D a)" 1 , (A o)~',and 

 (E a)~' and their repetitions. 



These some forms may be extended, as follows : Let E and E 

 severally denote the operations of changing a- into x + 1 and y into 

 // + 1 ; and let D^, D , A^, A y , be similarly interpreted with respect to 

 the differentiations and differences. We have then 



in which the function first operated upon is left out to save room. 

 Here and u may be either positive or negative integers. And even 

 i'-, or ft may be symbols of operation, but not with respect to x or y. 

 Thus 



(D - D )" 



in which the second side is to be thus interpreted. Changing y into 

 y ax, differentiate m times witli respect to .c, and then change y into 

 y + ax. 



We shall now give the heads of some methods of solution, observing 

 that this article is intended only for those who can already master the 

 same solutions by other methods. 



Take the common linear equation 



d"y d 1 // 

 jTT + o j___i + 



= V 

 *l 



in which a and 6 are constants, and x a function of .c. The operation 

 prformed upon y is OD" + iD-' + . . . : if this be called o, then y 

 is the result of performing the inverse operation c~' upon x. By 

 the method explained in FRACTIOUS, DECOMPOSITION OF, transform 

 (OD + 6D"-' +. . .)-' into A(D )-' +B(D -0)-' +,&c., where o, 

 IS, &c. are the roots of the algebraical equation ax" + Itx- 1 + . . . = 0. 

 Then y is 



A(D o)-' x + B(D - )-' x + . . . 



or \t~t-" xdx + 



substituting for D~ I its usual mode of expression. The arbitrary parts 

 of the solutions will be obtained by the constants of integration in the 

 usual manner. But the arbitrary part may always be obtained, in all 

 inverse operations, by considering the function operated upon as x + 0, 

 and operating separately upon x and 0. Thus (D a) ~ 3 x may be com- 

 pletely expressed by 



the second term of which is f (r + Q# + B* 2 ), P, Q, and B being any 

 constants. 



Suppose that theru are equal roots in the above equation, say 

 three roots equal to a. The resolution of the fraction gives terms 

 of the form 



K(D o)- 3 + L(D o)~ s + M(D - a)" 1 , 



which contribute to the general value of y, 



"{K( fdx? 

 and the arbitrary part " ( 



