CHAPTER XI. SUPPLEMENTAL ARTICLES. In the preceding Chapters the forms of quasi-constants have been determined by comparison of a proposed equation with general formulæ ; and this has been found sufficient for the majority of equations generally proposed for integration in the ordinary Works upon this subject. But the principle on which the Method of this Essay is founded is embodied in the right use of the equation Dudu + Dx. d ̧u+ D¡y.d ̧u+. or, more generally, ... Du D,t. Du +D,x.d ̧u+D,y.d1u+... = The use of other formulæ is a convenience in most cases, but not a necessity: and there are equations and classes of equations, the successful integration of which will depend solely on a skilful use of the above fundamental form whereby the change of the system of independent variables is effected. To the skilful use of this form we must look for an extension of success into fresh fields of difficulties. An example or two will illustrate what is here meant. None of our formulæ are of this form, we therefore arbitrarily assume that the new system of variables is such as to allow us to write The reason of the minus sign being assumed in this equation will be seen in the next step,-it gives us an integrable equation. Substituting u for § and eliminating v between equations (1) and (2), we have the integral required. The above is applicable to two independent variables only, but the following example is so stated as to be applicable to any number of independent variables, Du being equal to dru+ D1x.d ̧u+D1y.du +...; and the quantity represented by Qis under no restriction but this, that DQ is to be either a constant or a quasi-constant. Divide this by v and differentiate with D, remembering Divide by JDQ+v2 and integrate with S.; + S, (D1Q + v2) ̄`1 D, (Þ2o) = F (§, n, 5, ...) ; We must substitute for E, n, ... their values and then eliminate between this and equation (1); and the result will be the required integral. The following are particular examples of this class. (3) To integrate ht + kx + u (du + ad ̧u) = $ (du + ad ̧u). (4) To integrate х ht + kx+u (du + 2 d ̧u) = ¢ (du+2 d ̧u). (5) To integrate х ht + kx+ly + u (du + — d ̧u+ ad ̧u) = $(d ̧u + d ̧u+ad ̧u). t t II. The chief object in Chap. VIII. was to shew how to deprive a particular class of linear equations of coefficients which were not constant, in order to reduce them to linear equations with constant coefficients. These being always integrable, we were enabled to ascend from their integrals to those of the proposed equations. But in Chap. VII. we were often conducted for the value of u to a result of one of the forms following: (pdı)". u = 0, or (pdı)"t (pd.)1 ̄u = 0. It is desirable to exhibit the direct dependence of u on v; v being an auxiliary quantity defined by the equation that is, so will represent a series in integer powers of t from O to n-1, the coefficients of which are all arbitrary. We shall therefore take s0 to represent any series of the above form when all its coefficients are arbitrary. It is to be. distinctly understood that the series represented by s70 is definite in form and indefinite in the coefficients of its terms. This will give rise to a peculiarity in its use,-it will enable us with proper explanations to use the sign of equality in some cases where the strict logic of the steps would require the use of a sign of equivalence only. Since 80 is constant, by multiplying equation (1) by it we have Also, since 820 contains t to the first power only, if we operate with (pd) on vs20, by formula (1) Art. 39 we have (pd1) ⋅ vs¿10 = s;30. (pdı) v + s̟¿0. (¿'dı) v holds good for n = 1, and n=2. 2. We see also that it holds good in these cases because v satisfies equation (1); and that in consequence of this we may write for v any quantity whatever (or the sum of any such quantities) which satisfies equation (1). The form (3) will therefore hold good for n=1, 2, if we write for v any quantity such as (p'dı) v, or (p′′dı) v, because each of these when written for v in equation (1) will satisfy it as well as v. ...... We will now suppose that equation (3) is true for all integer values of n from 1 to (n-1); and we shall shew that in that case it is true for the next value of n. By the formula of Art. 36 we have 1 (pd.).vs.”0 = s¿^0. (pd.) v + s.” ̄10. (p'd;)v+ St 1-20. (p′′di) v + ... 1.2 |