difficulty. But though some general principles might be stated, the subject is best studied in the concrete application. In applying the above solution to the problem of attraction it is required to determine the arbitrary functions so that when r=0 we should have u = F(z). Now, since, when r = 0, log r is infinite, it is necessary to suppose (z) = 0. We have then 9. Equations whose symbolical form is binomial generally admit of solution by definite integrals. Pfaff's equation has thus been treated by Euler. (Lacroix, Tom. III. p. 529.) The very beautiful theorem of Parseval, which makes the limit of the series AA' + BB' + CC′ + &c. dependent upon the limits B C' of the series A+ Bu+ Cu+ &c. and A'+ + + &c., u u2 should be noticed. Suppose that, for all values of u, real and imaginary, Then, multiplying the equations together, AA' + BB'+ CC' + ... +Σ (a mum +! Bm) = p (u) y (u). u Assume, in succession, u=√1) and u= 67°√(1), and add the results, We find 2 (4A +BB+CƠ + ...)+2E (am cos me) +22 (3m cos me) = $ {e®√(−1)} & {e®√(~1)} + $ {e ̄®√(1)} & {e ̄°√(−1)}. = Now multiply by de, integrate between the limits 0 and π, observing that 2π, then AA' + BB'+. π [" (cos me) de = 0, and divide the result by 0 ... which is the theorem in question. Solution of Differential Equations by Fourier's Theorem. 10. As Fourier's theorem affords the only general method known for the solution of partial differential equations with more than two independent variables (and such are the equations upon which many of the most important problems of mathematical physics depend), we deem it proper to explain at least the principle of this application, referring the reader for a fuller account of it to two memoirs by Cauchy*. As a particular example, let us consider the equation Let u= (x, y, z, t) represent any solution of this equation. By a well-known form of Fourier's theorem, * Sur l'Intégration d'Equations Linéaires., Exercices d'Analyse et de Physique Mathématique, Tom. I. p. 53. Sur la Transformation et la Réduction des Intégrales Générales d'un Syatème d'Equations Linéaires aux différences partielles. Ibid. p. 178. successive applications of which enable us to give to u the form 1 u = 8773 81 4√(-1) p (a, b, c, t) da db dc dλdpdv......(38), vi-1) (a, b, c, 1) da do doi duc,....(38), ф where A = (α — x) λ + (b − y) μ + (c − 2) v. Substituting this expression in (37), and observing that from the form given to A we have d2 d2 d2 + + €1√(−1) · ¤a1√(−1) (— X2 — μ3 — v2), dx2 + dy dz2) = , + h2 (x2 + μ2 + v3) { $da db dc dλdpdv = 0, o being put for (a, b, c, t). This equation will be satisfied if o be determined so as to satisfy the equation Hence, integrating and introducing arbitrary functions of a, b, c in the place of arbitrary constants, we have the particular integrals, φ · = Bht √(−1) ↓2 (a, b, c), $= €-Bht √(−1) X1 (a, b, c)... (39), 1 Substituting the first of these values in (38), and merging 1 the factor in the arbitrary function, we have u = 877-3 SSSSSS e(4+ Brt) √(−1), (a, b, c) da db de dλdu dv... (40), a particular integral of the proposed equation. It may easily be shewn that the employment of the second value of ☀ given in (39) would only lead to an equivalent result. To complete the solution, we observe that if, representing da ď d2 dx2 + dy2+ dz2 by H, we make te, so as to reduce the given equation to the symbolical form, then, by Propositions II. and III. Chap. XVII., the transforma tion u = €o = dv dv de dt' will give H V €20 v = 0, D(D −1) which is of the same form as the equation for u. Hence, v admitting of expression in the form (40), we have, on merely changing the arbitrary function, E(A+Bht) √(−1) ↓2 (a, b, c) da db dc dλdμ dv ..... (41). The complete integral is thus expressed by the sum of the particular integrals (40) and (41). The sextuple integral by which the above particular values of u are expressed admits of reduction to a double integral leading to a form of solution originally obtained by Poisson. Cauchy effects this reduction by a trigonometrical transformation. It may be accomplished, and perhaps better, by other means; but this is a matter of detail which does not concern the principle of the solution. We may add, that when the function to be integrated becomes infinite within the limits, Cauchy's method of residues should be employed, The reduced integral in its trigonometrical form, together with Poisson's method of solution, which is entirely special, will be found in Gregory's Examples, p. 504. Cauchy's method is directly applicable to equations with second members, and to systems of equations. The above example belongs to the general form where H is a function of d d d dx' dy' dz For all such equations the method furnishes directly a solution expressed by sextuple integrals, which are reducible to double integrals if H is homogeneous and of the second degree. In the above example the double integration proves to be, in effect, an integration extended over the surface of a sphere whose radius increases uniformly with the time. Integrals of this class are peculiarly appropriate for the expression of those physical effects which are propagated through an elastic medium, and leave no trace behind. MISCELLANEOUS EXERCISES. 1. The complete integral of the equation is expressible in = the form u A+ Beh, A and B being series which are finite when n is an integer. (Tortolini, Vol. v. p. 161.) 2. The definite integral ["cos {n (0-a sin 0)) de, can be evaluated when n=+ (i + 1), where i is a positive integer or 0. (Liouville, Journal, Tom. VI. p. 36.) Representing the definite integral by u, it will be found that u satisfies The subject of the evaluation of definite integrals by the solution of differential equations has been treated with great generality by Mr Russell (Philosophical Transactions for 1855). 3. If v = a be the equation of a system of curves, v being dev d'v dx2 + dy3 = 0, a function of x and y which satisfies the equation and if u =ẞ be the equation of the orthogonal trajectories of the system, then u may be found by the integration of an B. D. E, 31 |