or r. Multiplying the above equations by dx, dy, dz respectively, and integrating, we have of which it is evident that two only are independent. Inte grating these, we have Squaring the last three equations and adding, we obtain a result which may be expressed in the form 2 2 dx (22 + 3*+ 2") { (de)" + (dy)* + (de) } – ( « %/ dt dt dt dy dz +2 + y de + d) dt dt = C2 + C ̧2 + C ̧2 = A3, 1 or, by virtue of (b) and of the known value of r, 2r2 (R + B) − (r dr)2 = 4a. = A2. 2 √{2r2 (R+B) − A3} rdr 2 Again, it is evident that by means of (c) we can eliminate R from each equation of the system (a). For (c) gives Substituting which in the first of the given equations, we have in which we must substitute for its value, viz. To this expression it would be superfluous to annex an arbitrary constant before that substitution. For each of the second members of (ƒ), (g), (h) is expressible in the form Ccos (p+C), in which is already provided with an arbitrary constant. The solution is therefore expressed by means of (e) and (i), which determine r and the auxiliary & as functions of t, and by (f), (g), (h), which then enable us to express x, y, z as functions of t. As we have however made no attempt to preserve independence in the series of results, the constants will not be independent. If we add the squares of (f), (g), (h), we shall have 2 1 = (a ̧2 + a ̧2 + a ̧3) cos2 4+2 (a,b ̧ + a,b, + a,b,) sin & cos & + (b,2+b2+b ̧2) sin2 4, 1 which involves the relations among the constants 2 2 a2 + a2 + a ̧2 = 1, b‚2 + b2+b2 = 1, a‚b‚ ̧±à ̧b2+a ̧b ̧=0.....(k). The six constants in (ƒ), (g), (h), thus limited, supply the place of only three arbitrary constants, and there being three also involved in (e), the total number is six, as it ought to be. In the same way we may integrate the more general system 2 where R is a function of (+...+x). The results, which have no application in our astronomy, are of the form which the above analysis would suggest. Binet, to whom the method is due, has applied it to the problem of elliptic motion. (Liouville, Tom. II. p. 457.) For all practical ends the employment of polar co-ordinates, as explained in treatises on dynamics, is to be preferred. 12. The following example presents itself in a discussion by M. Liouville*, of a very interesting case of the problem of three bodies. Sur un cas particulier du Problème des trois corps. Journal de Mathé matiques, Tom. 1. 2nd series, p. 248. where, for brevity, a' is put for cos (at + b), y' for sin (at +b). If we transform the above equation by assuming ux' + vy' = U, uy' — vx' = V, And these equations being linear and with constant coefficients, may be integrated by the process of the previous section. 10. Given where = X+T1 ̄ ̄Y+ T ̧ ̄Z+T‚ ̄T ' X= ax + by + cz, Y = ax + by + c'z, Z=a"x+by+c′′z, and T, T1, T2, T, are functions of t. 11. What is the general form of the solution of a system of n simultaneous equations of the first order between n + 1 variables? 12. What number of constants will be involved in the solution of a system of three simultaneous equations of the first, second and fourth order respectively between four variables? where r = (x2 + y2+22), seven first integrals are obtained of which it is subsequently found that five only are independent. How many final integrals can hence be deduced without proceeding to another integration? |