The class of letters that are termed characteristics, and which represent operations, is divided into two species ; those which denote functions, as F 4', X, and those which signify that some alteration is to bc made, which depends on the nature of the function on hich it is to be executed : thus f (1 -I- x2) means a function of I + a-2. In ails case the form of the function selected for f is quite independent on I -I- ; but if we have d (I + 2-2) the effect of the operation denoted by (I, depends on the form of the function inclosed within the parenthesis: this is the case with the symbols Those of the former species are called functional cha racteristics ; and as it will be convenient to have some generic name for the latter, we shall appropriate to (hem that of Derivative Characteristics, since their own form depends on, and is derived frotn that which follow: them.
The first of these species is subject to the following rule : Every functional characteristic affects all symbols which follow it, just as if they constituted one letter.
Thus, if ax =a x +6 x2, andfx= I x2, Then, afx=a(I x2) = a (1 x2) b (1 x2} And f ..e= 1 (a .x)2 =1 (a x b x2)2 .
Two signs have been made use of to represent mul tiplication : the cross ( x ) and the dot (.); and the sim ple juxtaposition of two letters, if they are in Italic, means the same thing : this last method is preferable in all simple cascs, as a + x a c, x2y a There exists a difference of meaning between the two former of these methods of indicating multiplication and the latter, which it is necessary to explain, and to which it is convenient to adhere. The dot and the cross imply a kind of disjunctive multiplication, or that, when they are interposed between two quantities before which a derivative characteristic is placed, this symbol only applies to the first of the two : x a Ea I As, 6 Ex-1 x+ a which represents A If the dot or cross Nvere not used in this case, the mean -I- a EX ing Nvould be quite different, for A would signify the complete difference of the quantity x + a V 1.) Another common signification of the clot is, that when is placed immediately after a derivative characteristic, it implies that the derivation extends until some other dot occurs Nvhich may separate it from any succeeding :actors : This is sometimes denoted by brackets of va rious kinds, thus, du d. xv.
lettotes the differential of u relative to x, multiplied by :he complete differential of xv. If written thus, du d. xv dx du it would signify the complete differential of xv In the operations of the differential calculus, it is constantly required to express the second and higher powers of a differential, as (d x)2, (d x)2, (d x)n, whilst it has rarely been found necessary to express the differ entials of the simple powers d (x2), d (x2), d (x4) These two series differ in the situation of the parenthe ses they contain ; but as the quantities in the first recur perpetually, it was desirable that they should be con tracted bv omitting the parentheses. Universal usage has sanctioned this omission, and they are always writ ten thus, d x2, d x3, d xfi; and in order to remove the ambiguity which this might occasion, in the few cases where the quantities mentioned in the latter series oc cur, thcy are distinguished by means of the point thus, d. x2, d. x3, .. d. xfi.
The principle that parentheses may be omitted, if it can be done without introducing ambiguity, has been partly adopted in the manner of expressing circular functions ; a department of analysis in which the inat tention to all general principles of notation has been severely felt. Five methods have been generally em
ployed for expressing thc squares and higher powers of the sines and cosines of arcs.
(sin. 0)2, sin. 82, sin. 20, (sin 0)2, sin 02.
The first of these has nothing objectionable in it ex cept the dot, which is not merely useless, but interferes with another principle. The second omits the paren theses, in compliance with the principle we have just stated : like the former, it has the useless appendage of the dot. The third is by far thc most objectionable of any, and is completely at variance with strong analo gies. An index, in the situation in which it there oc curs, invariably denotes repetition of the symbol to which it is attached : in this case it would therefore mean sin. sin. 0, or the sine of the sine of the arc O. It is true that this notation may be defined to mean the square of the sine, or any other function of the sine. Although a dcfinition cannot be false, it may be improper ; and the impropriety may arise either from its inducing am biguity, or from its offending against received princi ples; both which objections occur in the present in stance. Besides, if sin 20 were allowed to signify the square of the sine of 0, how should we denote the se cond sinc of 0 ? It may safely he asserted, that if this notation for the square of the sine were admitted, no convenient method could be devised, which should not also infringe some general principle. Thus one infrac tion would becoine the ground of introducing another, and instead of possessing a philosophical language founded on general laws, the science would arrive at one with no regularity to assist the memory, and devoid of those strong analogies which facilitate the acquisition of all othcrs.The fifth method, (sin 0)2, is made use of by Arbogast, in his work on the Calculus of Deriva tions; an authority which is entitled to the greater weight, because, from the nature of that excellent work, and the powerful use which is made of the various no tations employed in it, it is highly improbable that the author should have selected this out of the many re ceived notations, without having well considered its propriety. It differs but little from the last method, the parentheses only being omitted, which gives sin 02 a. mode of expression sufficiently clear for all simple cases. lf, however, become a compound quantity, thc pa rentheses must again be introduced. This notation has been employed by AI. Lacroix, in his great work on the Integral Calculus. M. Gauss and Delambre have also adopted it ; and it is likewise used in the latest and most valuable work of Lagrange.* In order to connect together various symbols which compose part of an expression, and which are subject ed to the same operation, different species of vinculaf are employed. From the great variety of these which are used by the printer, a considerable latitude of choice is allowed. In genenal, all symbols collected under a vinculum are to be considered and operated upon, as if they formed a single symbol. The choice of parenthe ses, although a matter of minor importance, is not alto gether arbitrary. An example, in which propriety and impropriety are apparent in different modes of writing the same expression, will be sufficient without the aid of formal rules.