The equivalent of this in the most general form is 19-0 2-0 X - a z-c' x-a' y-6) Observe the geometrical meaning: (97) indicates that the normal to the surface is perpendicular to a straight line which passes through a given point (a, b, c), and therefore the surface is generated by a straight line which passes through the given point and moves according to a given law: and this is a property of conical surfaces, of which therefore (98) is the general equation, and the arbitrary functional symbol contained in it expresses the law of the director-curve. dx (100) mz— nynx – Izly - mx' then multiplying the numerators and denominators severally by X, Y, z, and adding; and again operating in the same way with 1, m, n; the sum of denominators in each case is zero : therefore the sum of the numerators must also vanish : consequently x dx + y dy +z dz = 0; ldx + m dy +n dz = 0; (101) :: x2 + y2 +z2 = (1, lx +my+nz = C2; (102) .: x2 + y2 + z2 = f (lx + my+nz); (103) or u = F (x2 + y2 +z2, lx + my + nz) = 0; either of which is the equation of a surface of revolution, and in which the origin is on the axis of revolution; and equation (99) implies that all the normals of the surface pass through the axis : also from (102), which are the equations of a sphere and a plane, it follows that all plane sections of the surface, which are perpendicular to the line whose direction-cosines are proportional to 1, m, n, are circles. 386.] The assumption made in Art. 384, by which (84) is assumed from (83), requires further elucidation; and that our notions may be definite, I shall consider it from a geometrical point of view. Suppose the integral equation to be that to a surface; then, from (81) and (82), it appears that the normal to the surface at a certain point is perpendicular to the line whose direction-cosines are proportional to the values which P, Q, R have at that point, and also to any line of which the element on the surface is ds, the projections on the coordinate axes of ds being dx, dy, dz; and combining these two conditions, as in (83), it follows that the normal to the surface is coincident in direction with the normal to the plane containing these two lines (P, Q, R), (dx, dy, dz). Now the direction (P, Q, R) is fixed for any one point, and the direction of ds is indeterminate; in order therefore that we may leave the most general condition to be fulfilled hereafter, we may suppose these two directions to be the same, which fact is expressed mathematically by the equations dus (du, duzme (84): so that now that now (dxl dul (dz) are indeterminate are indeterminate, as appears from (83), and therefore the normal is only limited to being in the plane which passes through the point under consideration, and is normal to the line (P, Q, R). Thus far it appears that two consecutive points on the line (P, Q, R) will be on the surface, but nothing is determined as to consecutive points in other directions. Now suppose the integrals of the two equations (84) to be found, and to be (85): these are manifestly the equations to two surfaces, and, being simultaneous, express a line which is their line of intersection, and lies on the surfaces, and it is for all points along it that equations (84) are satisfied. The forms of these surfaces depend on the forms of P, Q, R; and as the equation of each of them contains an arbitrary constant, c, or C2, so by the variations of these, systems of surfaces arise, and by a relation which is arbitrary, but which we may assume to exist between these constants, we obtain a series of lines, all of which lie on the surface u = 0, and therefore by which, in their several and successive positions, the surface is formed; and this relation between c, and cy may be expressed by a functional symbol which will enter into the final equation; and although this function may be arbitrary, yet for any one surface it will be determinate; and hence will the values of G)GO), G) become determinate, and the position of the points contiguous to (x, y, z) be fixed in other directions than along (P, Q, R); that is, in other words, the resulting equations will express a continuous and determinate surface. Although then the assumption of (84) may appear to restrict the generality of (81), inasmuch as it causes the conditions expressed by it and (82) to be satisfied along only a line on the surface, yet it leaves us free to introduce the general functional symbol of relation between C, and ca, and thereby are we enabled to express the class of surfaces of the greatest extent which satisfies the condition of the given partial differential equation. The reader will perceive the agreement between the method here explained and the process of solution applied to the examples of the preceding Article. 387.] A similar method may also be applied to the integration of partial differential equations of the first order and first degree of any number of variables. Let the partial differential equation involve n variables, X1, X2, ... Im; and let us suppose the required integral to be of the form where all the variables are supposed to be independent; for if such were not the case, but if one were supposed to be a function of the other (n-1), the equation might be changed into the form (108) by means of equivalents analogous to (80). Now the total differential of (107) is and let us assume that the following (n-1) relations exist between (108) and (109), dify = db2 = ... = db. = u (say). (110) P, ** P. Me kvay! (1 ) Suppose now that we can determine the integrals of the (n-1) different equations which are involved in (110), or can by any means, as in Ex. 3 of Article 385, determine (n-1) different relations between the n variables ; and suppose them to be of the forms, fi (X1, X2, ... Xo) = (1, 12(X1, X2, ... 2n) = (2, fr-1 (X1, X2, ... Xy) = Cn-1; (111) where Cij Ca... Cn-1 represent (n-1) arbitrary constants. Then these arbury constants must be related to each other by a functiona mbol, such as • (C1, C2, ... Cn-1) = 0, (112) • (f1, 82, ... fn-1) = 0; (113) where .. are used as abbreviations for fi (X, X,, ... Xm), ... in (111) may thus be shewn: let (111) be differentiated, and we have and adding, the coefficients of dx,, dx , ... dxn in the sum are evidently (auto) .), ... Co ); and thus we have Note ) dx + ( a ) dx, +...+ (.) da, = 0; (115) and replacing dx, dxz, ... dx, by their proportionals from (110), wanase pile) + P3 (..) + ... + P. ( en ) = 0; (116) we have comparing which with (108) it is manifest that, with the excep- (117) (118) 388.] Also if we operate on the several equations of (114) with the series of equalities (110), by comparing the results with (108) it will be manifest that the functions f1, f2, ... fn-1 are all such as when substituted for u satisfy (108); and are therefore solutions of the given equation : each however will be less general than (117), because (117) combines them all under one other arbitrary functional symbol. I may however mention that although I have shewn that (117) is such as to satisfy the given equation, yet I have not proved that it is the necessary solution the question is, are any and what restrictions introduced by t hypothetical assumptions (110)? But these inquiries are on the range and scope of the present work. |