x x2 аз x3 Ex. 3. To develope{40+41 + as 12+ as 28 + ... f(ao) = a" ; [mam-1 dao] = ma。m-1a1, 1.2.3 [m (m-1) am-2 a1 dao+mam-1 da1] = m (m −1) am-2 a12 + mam-1 [d2 f(ao)] = m-2 m = aTM+maTM-1a1 = + maTM¬2 { (m −1 ) a ̧2 + A© A2} x2 1.2 23 +mao" + which is the Multinomial Theorem; and in which it is to be observed, that a, does not appear in any coefficient before that of x. Ex. 4. To develope e*, where ... [e a1 dao+e da1] = eaa12+eɑ2, [d3 f(ao)] =eo {a3+3α1 α2+αз}, and the student is recommended to practise himself in these and other examples, as the process is simple and very useful. The preceding process is, from another point of view, an extension of Taylor's Series; for whereas by that we are able to expand f(a+x), by this we can develope in ascending powers of x, ƒ (a+a1 1 + ...) ; and therefore if we put a2 = ɑ3 x2 1.2 = ... = 0, the preceding formulæ become those of Taylor's Theorem. 96.] Sometimes, instead of finding the actual coefficients, it is convenient to find the law of their dependence; that is, to determine the equation by means of which they are related to each other; such as has been found by implicit differentiation in Art. 87, equation (126). Thus in Ex. 1 of the preceding Article, let where C1, C2, C3, ... are the coefficients of the powers of a, arising from the product of the developed values of y and z, and which, as determined above, are the several values of d.soαo, d2. Ao αo, d3. Ao αo, ... on the supposition that da, a1, da1 = a2, ... dao = A1, dA1 A2, dAg A3, ...; but, on comparing (164) and (165), it appears, that = = And therefore by Leibnitz's Theorem, Art. 55, equation (4), whence A1, A2, ... may be successively calculated. As an example of this process, consider the problem which was discussed in Art. 87, viz. the expansion of Here we have merely to replace in (166) αo, ɑ1, ɑ2, and we have 1 1 by 1, 1 n (n-1) An+ 2 N An-1 + 3 1.2 by means of which the successive coefficients can easily be calculated. and by a similar process the other coefficients may be found; and these are severally the numbers of Bernoulli. 97.] Again, if u = f(a, ß, y, ...), where where a。 = f(αo, bo, co,...), and An = d"ƒ(ao, bo, co, ...); the total differentials being taken, and on the supposition, as heretofore, of daa1, da1 = a, ...; db。 b1, db1 = b2, ...; dc = c1, = Also since do Ao bo, we may by successive differentiation, as in the last example, implicitly determine the relation between the successive coefficients. As a particular example of the last theorem, suppose and A1, A2, ...... u = Aq + A1 + A2 x2 + 1.2 A0 = f(ao, a1, a2, ......), are the successive total differentials of A, on the supposition that da。 = a1, da1 = A2, The process which is developed and applied in the preceding Articles was first communicated to me by Mr. W. F. Donkin, the Savilian Professor of Astronomy at Oxford. It is somewhat similar to that which is the subject of Arbogast's Calculus of Derivations. The form of the series which is here taken as the subject of the functional symbol is different to that chosen by Arbogast; and the method by which I have shewn that the differentials of the several constants will in the expansion be replaced by their next following constants is also different. It is however convenient to call the process by the name which he has given to it, although the preceding form of it is due to Professor Donkin. The student also may consult the Treatise on the Differential and Integral Calculus, by Augustus De Morgan, London, 1842, Art. 214-227; the large treatise on the Differential Calculus, by S. F. Lacroix, Paris, 1810; and papers in the Cambridge and Dublin Mathematical Journal, by Mr. De Morgan, vol. i. p. 238 and vol. vi. p. 35, and by Professor Donkin, vol. vi. p. 141. To complete the theory of the expansion of functions in ascending powers of the subject-variables, it is necessary to inquire into the form which the general or nth term of such an expansion assumes: because on the form of this and its relation to the preceding and the succeeding terms does the convergency or the divergency of the series depend. But this requires conditions which at present we are unable to assign: and therefore the remainder of our investigations on the subject must be reserved until the sixth Chapter. SECTION 7.-On the formation of Differential Equations by the elimination of constants and functions by differentiation. 98.] An equation which contains differentials or derivedfunctions is called a Differential Equation; and it is called an equation of the first, second, ... nth order, according to the order of the differential or derived-function involved in it. It is also called a partial or a total differential equation, according as the differentials which it contains are partial or total. Thus equations (56), (62) of Art. 52 are partial differential equations of the first order; Ex. 1 and 2 of Art. 77 are total differential equations of the second order. |