We find 2 (AA' + BB + CC′ + ...) + 2Σ (1⁄4m cos me) + 2Σ (8m cos mỹ) = $ {e®√(−1)} & {e®√(1)} + $ {e ̃ ̄°√(1)} & {€ ̄°√(−1)}. Now multiply by de, integrate between the limits 0 and π, observing that [* (cos me) de = 0, and divide the result by 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. 1. p. 53. Sur la Transformation et la Réduction des Intégrales Générales d'un Système d'Equations Linéaires aux différences partielles. Ibid. p. 178. successive applications of which enable us to give to u the form 1 u = SSSSSS '√(−1) † (a, b, c, t) da db dcdλdμdv......(38), where A (ax) λ + (b − y) μ + (c − z) v. = Substituting this expression in (37), and observing that from the form given to A we have 1 877-3 81 + b2 (x2 + μ2 + v2)} þđɑ db đc dλdμdv=0, being put for (a, b, c, t). This equation will be satisfied ifo 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, $ = €3he√(−1) ¥, (a, b, c), where B = (x2+μ2 + v2)3. -Bht√(-1), $==√(1)x, (a, b, c) ... (39), φ=ε Substituting the first of these values in (38), and merging 1 the factor in the arbitrary function, we have นะ =SSSSSS ((1+BM()√(−1) of, (a, b, c) da db dc dλ dpdv ..... (40), 1 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 7 d2 dx*+ 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 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, 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 the method furnishes directly a solution expressed by sextuple integrals, which are reducible to double integrals if His 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. -hx is expressible in the form u = A + Be ̃ ̄*, A and B being series which are finite when n is an integer. (Tortolini, Vol. v. p. 161.) 2. The definite integral ["cos {n(8-x sin 6)} 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 an equation of the form d2u = A+ 26. d.c The subject of the evaluation of definite integrals by the solution of dif ferential equations has been treated with great generality by Mr Russell (Philosophical Transactions for 1855). 3. If va be the equation of a system of curves, v being a function of x and y which satisfies the equation d2v d2v dx dy 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 exact differential equation of the first order, and when found d'u d2u will satisfy the equation = 0. dx2 + dy The above theorem is applied by Professor Thomson to the problem of determining the forms of the rings and brushes in the spectra produced by biaxal crystals. (Cambridge Journal, 2nd Series, Vol. 1. p. 124.) 4. The normal at a point P of a plane curve meets the axis in G, and the locus of the middle point of PG is the parabola y=lx. Find the equation to the curve, supposing it to pass through the origin. (Cambridge Problems.) 5. The normal at any point of a surface passes through the line represented by 2. Find the differential = m n equation to the surface, and obtain the general integral. (Ib.) 6. Prove that the differential equation of the surfaces generated by a straight line which passes through the axis of z, and through a given curve, and which makes a constant angle with the axis of z, is 7. Integrate the above equation. 8. Express by a definite integral the series, Form the differential equation by Chap. XVII. Art. 11, and then apply Cambridge Transactions, Vol. ix. p. 182.) 9. Hence express the series in a form suitable for calculation when x is large. Proceeding according to the directions of Chap. XVIII. the complete integral of the differential equation expressed by descending series will be u=x ̄ {(4 cos x + B sin x) R + (A sin x − B cos x) S}, |