"eme u dOn = emB + m) "u, (35) id \~" C" \j0+m) u - e-m0j emeudO"; (36) and if F expresses an algebraical function, r(jL)em*=emev(jL+m). (87) From these expressions another theorem may be deduced. Since from (32) we have \de' ~ do" u; (38) in it let us substitute for To successively 0, —1, —2,... — (n —1); and let all these processes be performed successively on u; then we have (»-')£• _c doe doe dou = e*»(e-»jL)\; (39) which and (30) and (37) are three fundamental theorems of this kind of operating symbol. 423.] And they may by the following substitution take forms which are useful in the solution of many differential equations. 424.] By this last formula, differential equations, of which / d \n the differential terms are of the form [x u, that is, where x is equicrescent, and the power of x in the coefficient and the order of the derived-function are the same, may be transformed into others of which the coefficients shall be constant. Because, if x = es, (•#.-(£-.+')(s--+»)-"(£-0i* <«> Of this transformation some examples are subjoined. Ex.l. = log*. By (40) this becomes, if x = ee, d .\ d du (57) In a paper by Mr. Hargreave in the Philosophical Transactions for 1848, (51) and (57) are extended to algebraical functions of -r- ■ ax 127.] Taylor's series may be expressed in the following concise form, if the symbol of quantity is separated from that of operation. By (76), Art. 71, if we replace derived functions by their equivalent ratios of differentials, we have /(. + *> = /(*) + J£l T + -£> „ + (58) and therefore *, M 51 d h Id \% h* td V h* \ e, ^ /(•+*) - {i + B j + (s) o + (a) 133 + »■ \M = eh3i/(x); (59) » — so that if f(x) is operated upon by the symbol e <tr, it is changed into f(x + h). Similarly, if f(y) is operated upon by the symbol e du, it is changed into f(y + k). And therefore if r(x,y) is a function of two independent variables x and y, ehT* +*SF F (x, y) = F (x + h,y + k); (60) PRICE, VOL. I. 4 1 428.] Since by Euler's Theorem, Art. 82, if u is a homogeneous function of n dimensions, and therefore the effect on u of the operation symbolized by x -3- + y ^j- + ... is to convert it into nu: and therefore But ou these subjects I must say no more: my work has already well nigh exceeded the limits required in a didactic treatise, and many theorems and processes have been omitted, not because they are useless or inelegant, but because I could not afford the space. It is however the less necessary to enlarge on this calculus of operations because Mr. Carmichael, Fellow of Trinity College, Dublin, has lately published a treatise on the subject*, to which I am indebted for reference to some subjects in the preceding pages. I cannot however conclude without recommending the student to consult (1) " Essai sur un nouveau mode d'exposition des Principes de Calcul Differentiel," by M. Servois, Nismes, 1814; (2) many papers in the Cambridge Mathematical Journal by Mr. Gregory, Professor Donkin, Professor Boole, Mr. Bronwin; (3) the memoir of Professor Boole, "On a general method of Analysis," in the Philosophical Transactions for 1844; (4) a memoir of Mr. Hargreave in the Philosophical Transactions for 1848; and (5) the papers of M. Servois in the Annales des Mathematiques de M. Gergonne, Vol.V. * A Treatise on the Calculus of Operations, by the Rev. Robert Carmichael, A.M.; London, Longman and Co., 1855. END OF VOL. I. |