Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/38

28 INFINITESIMAL CALCULUS The preceding method can be readily extended to the general case in which the equation of the moving curve contains any number n of variable parameters, which are connected by nl equations of condition. 82. The theory of envelopes, or of ultimate intersections, may be said to have originated with the investigations of Huygens on evolutes, already referred to, and those of Tschirnhausen on caustics (Ada Enulitorum, 1682). These authors, however, merely treated geometrically a few cases of moving right lines, and did not give any general method for the investigation of such problems. Leibnitz was the first who gave a general process for the solution of this class of questions (Ada Eruditorum, 1692, 1694). His method does not differ in any material respect from that here given. (1) To find the envelope of the parabolas described by a pro jectile discharged from a given point with a given velocity, but at different angles of elevation. If e be the angle of elevation, and h the height due to the initial velocity, the equation of the parabolic path is x => ?/ tan c =-= TT 4/4 cos-e Let tan c = o, and the equation becomes Consequently the equation to the required envelope is which represents a parabola. This problem is the first that was brought forward on the locus of the ultimate intersection of curved lines. It was proposed by Duillier to John Bernoulli, who solved it, but not by any general method (Commcr. Epist. Lcib. ctBcrn., vol. i. p. 17). (2) To find the envelope of the system of conies represented by the equation where a is a variable parameter. Proceeding as before we get as the equation to the envelope (x/iif + y 2 = Q. Hence we infer that a system of confocal conies may be regarded as inscribed in the same imaginary quadrilateral. (3) Find the envelope of the plane x 11 z ^- + + = 1 I m n in which the parameters I, m, n are connected by the equation lmn = a*. Ans. %7xyz = a 3 . (4) A right line revolves with a uniform angular velocity, while one of its points moves uniformly along a fixed right line, prove that its envelope is a cycloid. Symbolic Methods. 83. The analogy between successive differentiation and ordinary exponentials was perceived by Leibnitz and the early writers on the calculus, and afterwards more especially by Lagrange (Mem. Acad. Berlin, 1772). Arbogast was, however, the first to separate the symbol of operation from that of quantity in a differential equation (Calcul dcs Derivations, 1800). The first writers who appear to have given correct rules on the subject of operations were Francois, Ann. des Math., 1812, and Servois, in the same journal, in 1814. Servois more especially exhibited the principles on which the legitimacy of the separation of the symbols of operation from those of quantity depends ; and, making a separate calculus of functions out of those properties, he succeeded in proving that differences, differentiations, and multiplications by any factors which are independent of the variable, may be employed as if the symbols of operation were ordinary algebraic quantities. Hence has arisen a new method of considering the principles and processes of the calculus, called the symbolic method, or the calculus of operations. In this method ~ is written in the form [ u. and the symbol dx dx) -j- is regarded in the light of an operation, supposed to be made on the function u according to the established principles of differentiation- Ag!lin Also, dxj dx) dx ) u . . . . (3). And, if u be a function of x and y, d_ dx_ Hence we observe that the symbols -- and - - operate and arc com- dx dy bined according to the same laws as ordinary algebraic symbols of quantity, such as a and b ; and we can readily infer that the theorems in ordinary algebra (compare ALGEBRA, vol. i. p. 519, 8, 9) which depend solely on such laws of combination are capable of being extended to similar theorems depending on the symbols and , dx dy or on the symbol and any constant a. Such results are in general capable of extension to any symbols that are subject to the same laws of combination. The law embodied in equation (1) is called the distributive law; the second, in (2), is called the index or exponential law ; and the third, in (3), the commutative law. It is convenient to denote the preceding symbols by single letters. Accordingly we may suppose the symbol to be represented by I), and by D , &c. dy In general, if ir, p denote two symbols of operation such that TT(U + v) = irU + irV , p(u + v) = pu + pv , irpu = pirU , ir m ir n u Tr m +"U , then the symbols ir, p possess the distributive, commutative, and exponential properties. For example, suppose EA represent the operation of changing x into x + h in any function of x, i.e., suppose Ek<p(x) = <p(x + h). Then EA | <p(x = <p(x Moreover, Ex- denoting the operation of changing x into x + k, we have . . EA . ~ = Eh<t>(x] ging + k) = <p(x + h + k) . In like manner Hence the symbols EA, Ek are commutative. Also the equation may be written, symbolically, thus : This shows that the symbol EA is of the nature of an exponential ; and may be written in the form E h . 84. This symbol can also be connected with Taylor s expansion. Thus, if we separate the symbols of operation from those of quantity in Taylor s theorem, it may be written Accordingly, although we can give no direct meaning to the symbol e* D , except as the representative of the symbolic expansion we may from the preceding section regard it as equivalent to the symbol E h . In like manner we may write e hv <t>(x, y) = 4>(x + h,y). If now we suppose both sides operated on by the symbol c ; D , we have e* D . e kD . <p(x, y) = e kD (f>(x + h, y) = <p(x + h, y + k). Hence V>(z, y) { 1 + (AD + 4DO + |(/4D + D ) 2 + . . . } (x, y) (Compare 57 ; also Arbogast, Gal. des Der., pp. 343-352.) 85. Another proof, by the method of operations, of the foregoing symbolic expression for Taylor s theorem may be added. It has already been shown that when h is infinitely small we may write In like manner (1 + A And in general Now suppose nh = a, an! we get