it is evident that, in order that it should be derivable from a single primitive, we must have Q = P R = Q P R (2) +- (4) F. (4) +- (2); (d) f = (4) F T' T dz T' dz T dx T' T d where refers to a not only as appearing independently, but also as implicitly involved in t; and so on for the rest. which are the equations of condition of existence of a single complete primitive. It is evident from the symmetry of the problem that the must also hold here. But this is not a new condition. It may be deduced from (26), by multiplying the respective equations of that system by R, P, and Q, and adding the results, It is obvious that when there exist n variables, the num(n-1) (n-2) ber of independent equations of condition is being the number of ways of equating two partial differential coefficients in a system in which n−1 are contained. 2 The solution of any such equation may be effected by an extension of the method adopted for equations with three variables. We must integrate as if all but two of the variables were constant, adding, in the place of an arbitrary constant, an arbitrary function of the variables which remain. This function we must determine by differentiating with respect to all the variables, and comparing with the equation given. If a single primitive exist, such determination will be possible. If a single primitive do not exist, we must, following the analogy of the corresponding case for three variables, endeavour to express the solution by a system of simultaneous equations. And such is indeed its general form. Pfaff, in a memoir published by the Berlin Academy 1814-15, has shewn that, according as the number of variables is 2n or 2n+1, the number of integral equations is n or n+1 at most. His method, which is remarkable, consists of alternate integrations and transformations. For important commentaries and additions see Jacobi (Werke, Tom. I. p. 140), and Raabe (Crelle, Tom. XIV. p. 123). Ex. Given (2x+y2+2xy,−y1) dx+2xydy-xdy,+x3dy2 = 0. If we suppose the variables y1, y, constant, we have to integrate 2 (2x + y2+ 2xy, −y1) dx + 2xydy = 0, which, on substituting an arbitrary function of y1, y2 represented by 4, for an arbitrary constant, gives Differentiating with respect to all the variables, we have (2x+y3 + 2xy, −y1) dx + 2xydy — xdy1+x3dy ̧ Had we begun by making x and y constant, we should have had as the result of the first integration, denoting a function of x and y. Differentiating with respect to all the variables and comparing with the given equation, we should find the substitution of which in (b) reproduces the former solution (a). Equations of an order higher than the first. 10. When an equation of the form Adx2 + Bdy2 + Cdz2 + 2Ddy dz + 2Edxdz + 2Fdx dy =0.......(28) is resolvable into two equations each of the form Pdx+Qdy + Rdz=0, the solution of either of these obtained by previous methods, will be a particular solution of (28), and the two solutions. taken disjunctively will constitute the complete solution, which is therefore expressed by the product of the equations of these solutions, each reduced to the form V=0. The condition under which (28) is resolvable as above, is expressed by the equation ABC+2DEF – AD2 - BE2 – CF2 = 0 ....................... (29). B. D. E 19 This is shewn by solving (28) with respect to do, and assuming the quantity under the radical to be a complete square. = 0, Thus, the equation x2dx2 + y2dy2 — z3dz2 + 2xydx dy which will be found to satisfy the above condition, is resolvable into the two equations Geometrically the solution is expressed by lines drawn in any manner on the surface, either of the sphere (a), or of the hyperboloid (b). When the condition (29) is not satisfied, the proposed equation does not admit of a single primitive, or of any disjunctive system of primitives. But it does in general admit of a solution expressed by a system of simultaneous equations. Thus, if we integrate the equation dz2 = m2 (dx2 + dy2), supposing a constant, we find z=my + C, or, replacing C by a function of x, On substitution and integration, we find that this will satisfy the proposed equation if we have the system (c) (d) will therefore constitute a solution of the equation given. We enter not into the question whether it is the most general solution or not, proposing merely to exemplify the kind of solution of which the equation admits. To this we may add that all equations which do not satisfy the conditions of integrability, though they may present themselves in the form of ordinary, have a far more intimate connexion with partial differential equations; and that this connexion affords the best clue to the solution of their theoretical difficulties. 1. † 2. 3. (x − 3y − z) dx + (2y − 3x) dy + (z − x) dz = 0. × (y + z) dx + (≈ + x) dy + (x + y) dz = 0. × · 4. yzdx + zxdy + xy dz=0.X 5. (y + z) dx + dy+dz = 0. X 6. ay31⁄23dx+b22x2dy + cx3y3dz = 0. X 7. (x3y—y3—y3z) dx + (xy2 — x2z — x3) dy+ (xy2+x3y) dz=0. × 8. (2x2+ 2xy + 2xz2 + 1) dx + dy + 2zdz = 0, x 9. (2x + y2+ 2xz) dx + 2xy dy - dw + x2 dz = 0. 10. Is the equation (1+2m) xdx + y (1 − x) dy + zdz = 0 derivable from a single primitive of the form (x, y, z) = c ? 11. Shew that any system of lines described on the surface of the sphere + y2+2= r2, and satisfying the above equation, would be projected on the plane xy in parabolas. 12. Shew that Monge's method would, if we integrate first with respect to x and z, present the solution of the equation of Ex. 10, in the form (1 + 2m) x2 + z2 = 4 (y), 2y (1 − x) = − p′ (y). 13. Applying this form to the problem of Ex. 11, form and solve the differential equation for the determination of (y), and shew that it leads to the result stated in that Example. 14. Find the equation of the projections of the same system of curves on the plane yz. |