425.] Let u and be two functions of x: then In the right-hand member of this equation, let & be the symbol of differentiation referring to u only and d' that referring to v only then d dx δ (48) (49) = + Let both members be raised to the nth power; then which is Leibnitz's Theorem given in Art. 55. Of this theorem the following are particular examples. let us take D, d, d' to be symbols of x-differentiation; and let us assume D to apply to both u and v, and ò and d to u and to v separately so that taking symbols of operation only Let both members of this equation be raised to the nth power; In a paper by Mr. Hargreave in the Philosophical Transactions for 1848, (51) and (57) are extended to algebraical func 427.] 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 h so that if f(x) is operated upon by the symbol ea, it is changed into f(x+h). d k Similarly, if f(y) is operated upon by the symbol edy, it is changed into f(y+k). And therefore if r (x, y) is a function of two independent variables x and y, 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 But on 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 Différentiel," 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 Mathématiques 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. |