Legendre, in his Exercises du Calcul Integral, ploys a different notation to express a function very nearly allied to that wc are considering. He denotes x — I .. 2.1 by 7 (x.) Euler also treated of the perties of the co-efficients of a binomial, and denoted 9 n them by the characters (7) (73), Eec. Of these various methods of denoting the same thing, that of Van dermonde seems to be the most advantageous. 'Ube reasons of this preference we shall shortly state. When x is repeated n times, x.x...x, it is represented by xn ; and generally, when any letter denoting either operation or quantity is repeated, it has the letter indicating the number of repetitions placed in the right hand corner above it ; such is the case in the following instances, xn, dn, fn. It i$ diet efore correct, as (lir as regards analogy, to place the index in that situation ; so far then as relates to the position of the letter n, both the nota tions of Vandermonde and that of Arbogast are equally admissible; this, howevee, is not the case with the letter a. It is used for the difference of two successive factors, and has no regard at all to repetition ; it is not therefote consistent with analogy to place it in a similar situation.
Another, and a very. powerful reason for prefetaing the system of Vandermonde is, that it admits of unlimited extension, without in the least infringing on typographi cal facility. When the factors have their second differ ence zcro, they are denoted thus : Lacroix has assigned to such expressions the appro priate name of powers of the second order. If the third differences are zero, they are called powers of the third order, and are readily denoted thus : and similarly for the higher orders : this notation there fore admits of a very easy extension to complicated cases, without increasing its difficulty of execution. Perhaps it may be urged that the notation of AI. Kratnp may be extended as follows : but the difficulties of the printer will be greatly increas ed the moment any substitutions are made for a or b, and the other objections remain the same.
These inconveniences attend the method of Arbogast in at least an equal degree. That which Euler has em ployed ;.s adapted to the particular purpose he has in. view ; but if it be allowed that sufficient reasons have been produced to justify the preference we have assigned to that of Vandermonde, it is highly desirable that it should be employed on all future occasions. We have observed that Al Legendre uses (x) to represent x x — 1 ... 2.1, whether x is a whole number or a fraction; this can scarcely be tcrmed a new notation ; but it is certainly rather to be regretted, that so able an inquirer should not have given his sanction to a notation previ ously established, and uhich agrees so well with those rules on which all notation should be founded.
The two signs which denote differentiation anti in tegration are, in strictness, liable to the objections of this rule ; and a similar remark is applicable to the similarly related signs A and In several of these instances, mathematical signs have becn needlessly multiplied by not attending to this prin ciple, that whenever 7VC wish to denote the inverse of any operation, 7ve must use the sante characteristic with the index — I. This principle being acknowledged, we shall in future be delivered from one cause of the redun dancies of signs. Should, for example, any future in quirer, when considering the calculus of iariations, have occasion for the inverse operation of that denoted by el`, lie must not, conformably to a hasty view of analogy, re present it by G-; but, directing his attention more deeply to the subject, he will perceive the necessity of indicating it by cri Again, in the equation x = tan y, thc value of y, in terms of x is usually expressed y = arc tan x ; this is both long and inelegant. It should, in complying with the rule just laid down, be written thus: y = tan x, and similat ly' with other circular functions; this method is likewise attended with the advantage, that all this class c)f functions will then be indicated by three letters.
Another principle, whose importance becomes eminent in proportion as our investigations become general, is, that every equation ought to be capable of indicating- a law. Signs must not be employed at random, nor must new ones be introduced without grave necessity. When, however, unusual combinations, or the demonstration of new properties, render such a rcsource indispensa ble, it becomes particularly desirable that those which are first contrived, should be possessed of such pro priety and power as shall effectually preclude future in novators from all temptation to change them. Analogy to those which form the established language of the science, although a ptecept of great importance, can not be admitted to supersede the rigid enforcement of that we arc now considering: happily, however, the two principles will rarely be found at variance. for those symbols, and those inflexions of symbols, (if the telm may be allowed,) which long experience has natura lized, generally furnish the most correct models of imi tation. A precept in some measure connected with this principle,although it may perhaps tic considered of mi nor importance, may be introduced.