Substituting this, with the corresponding value of q derived from (a), in the equation dz = pdx+qdy we have integrating which in the usual way, we find (1 − z 3) 3 — — cx − (1 − c )3y—c', √ or, changing the signs of c and c', 2 (1-2) = cx = cx − (1 − c2) y + c (c), or c2x2+ (1-c2) y2 + 22 - 2 c√ - c2 x. y. +2cc'x. -2c'√π-222y + c2 = 1 which is a complete primitive. The corresponding form of the general primitive will be But another system of solutions exists; for from the first, third, and fourth members of (b) we may deduce dx- pdz pdz+zdp + dx = 0, 221 zdp, pdz = z2 dx-dx = -z ^ whence pz + x = a, from which, and from the given equation determining p and q, we have to integrate α-x {1 − (a — x)2 — z2} 3 dz = dy. .(e), To deduce the singular solution from the differential equation (a) we have whence p=0, q=0; substituting which in (a) we find The above example illustrates the importance of obtaining, if possible, a choice of forms of the complete primitives. The second, of those above obtained, leads to the more interpretable results. It represents a sphere whose radius is unity and whose centre is in the plane x, y, while the derived general primitive represents the tubular surface generated by that sphere moving but not ceasing to obey the same conditions. The singular solution represents the two planes between which the motion would be confined. All these surfaces evidently satisfy the conditions of the problem. Ex. 2. Required to determine a system of surfaces such that the area of any portion shall be in a constant ratio (m: 1) to the area of its projection on the plane xy. The differential equation is evidently 1 + p2 + q2 = m2, Forsyth, p. 305, Ex. 3.I and it will readily be found that it has only one complete primitive, viz. z = ax+ √√(m2 — a2 — 1) y + b. Thus the general primitive is z = ax + √/(m2 — a2 − 1) y + $ (a), α 0=x √(m2 — a* − 1 ) Y + (a); and this represents various systems of cones and other developable surfaces. how? Similar but more interesting applications may be drawn from the problem of the determination of equally attracting surfaces. 12. Attention has already been directed to the different forms in which the solution of a non-linear equation may sometimes be presented. It may be added that linear equations admit generally of a duplex form of solution. The ordinary method gives directly the equation of the system of surfaces which they represent; Charpit's method leads to a form of solution which exhibits rather the mode of their genesis. Ex. Lagrange's method presents the solution of the equation of. p.330 ....(a), (mz-ny) p + (nx - lz) q=ly-mx ........... in the form lx+my+nz = $ (x2 + y2 + z3) .................. .(b), the known equation of surfaces of revolution whose axes pass through the origin of co-ordinates, Charpit's method presents as the complete primitive of (a) (x − cl)2 + (y — cm)2 + (z − cn)2 = r2.........................(©), ..... c and r being arbitrary constants. This is the equation of the generating sphere. The general primitive represents its system of possible envelopes. These solutions are manifestly equivalent, Symmetrical and more general solution of partial differential equations of the first order. 13. The method of Charpit labours under two defects. 1st, It supposes that from the given equation q can be expressed as a function of x, y, z, p; 2ndly, It throws little light of analogy on the solution of equations involving more than two independent variables-a subject of fundamental importance in connexion with the highest class of researches on Theoretical Dynamics. We propose to supply these defects, dz m2-ny dy It will have been noted that Charpit's method consists in determining p and q as functions of x, y, z, which render the equation dz=pdx + qdy integrable. This determination presupposes the existence of two algebraic equations between x, y, z, p, q; viz. 1st, the equation given, 2ndly, an equation obtained by integration and involving an arbitrary constant, Let us represent these equations by F (x, y, z, p, q) = 0, Þ (x, y, z, p, q) = a .......(29), respectively. And let us now endeavour to obtain in a general manner the relation between the functions Fand Þ, Simply differentiating with respect to x, y, z, p, q, and re dF presenting by X, by X', dF dp dp we have Xdx + Ydy + Zdz + Pdp + Qdq= 0, X'dx + Y'dy + Z'dz + P'dp + Q'dq = 0; or, substituting pdx + qdy for dz, (X+pZ) dx + (Y+qZ) dy + Pdp + Qdq= 0....(30), (X' + pZ') dx + (Y' + qZ') dy + P'dp + Q'dq = 0....(31). ✓ Substituting these values in (31) we have (X' + pZ' + rP' + s Q') dx + (Y'+qZ' + sP' + tQ') dy = 0, ✓ which, since dx and dy are independent, can only be satisfied by separately equating to 0 their coefficients. These furnish then the two equations Now these equations are of the same form as (32). They establish the same relations between the functions − (X′+pZ'), − (Y'+qZ'), P', Q',.........(34), as (32) does between the differentials dp, dq, dx, dy. It follows that if we give to dx and dy, which are arbitrary, the ratio of the last two of the functions (34) then will dp and dq have the ratio of the first two, so that the following will be a consistent scheme of relations, viz. Now dividing the successive terms of (30) by the successive members of (35) we have (X+pZ) P'+ (Y+qZ) Q' − P (X' +pZ') − Q (Y'+qZ') = 0......(36). between F This is the relation sought. It might be obtained by direct viz. the relation and I, see élimination by multiplying the equations of (33) by P and Q respectively, and the corresponding equations derived from top of p.357. (30) by P and Q respectively, and subtracting the sum of the former from the sum of the latter. It is obvious too, and the remark is important, that we might pass directly from (30) to (36) by substituting for da, dy, dp, dq, the functions of (34), and that this substitution is justified by the identity of relations established in (32) and (33). If in (36) we substitute for X, Y, &c. their values, and transpose the second and third terms, we have Such is the relation which connects the functions F and P. |