Realization of complet. ed solution. for the q's; with the same coefficients 11, 12, etc., given by the following formulas : Resultant motion reduced to motion of a conserva tive system with four fundamental periods and farther, that 11, 12, etc. are the negatives of the coefficients of a, a,a,, etc. in the quadratic } {(12 a ̧ + 13 a ̧ + 1 4 a ̧)2 + (21 a ̧ + 2 3 a ̧+ 24 a ̧)2 + etc.}... (51) expanded. Hence if G (aa) denote this quadratic, and G (pp), G (99) the same of the p's and the q's, we may write (47) and the corresponding equations for the q's as follows: equal two and two. Algebraic theorem. Details of realized solution. = 0, - n3q, + These equations are harmonized by, and as is easily seen, only by, assigning to n one or other of the two values of -λ given in (32), above. Hence their determinantal equation, a biquadratic in n2, has two pairs of equal real positive roots. We readily verify this by verifying that the square of the determinant of (42), with λ replaced by n', is equal to the determinant of (47) with 11, 12, etc. replaced by their values (48). Hence (§ 343g) there is for each root an indeterminacy in the ratios P/P1, P/P1, P/P1, according to which one of them may be assumed arbitrarily and the two others then determined by two of the equations (47); so that with two of the p's assumed arbitrarily the four are known: then the corresponding set of four q's is determined explicitly by the firsts of the pairs (46). Similarly the other root, n', of the determinantal equation gives another solution with two fresh arbitraries. Thus we have the complete solution of the four equations with its four arbitraries. The formulas (46)...(52) are clearly the same as we should have found if we had commenced with assuming 41 =p, sin nt + q, cos nt, 4,=p, sin nt + q, cos nt, etc....(54), as a particular solution of (53). Details of realized solution. 345. Important properties of the solution of (53) are found Orthogothus: (a) Multiply the firsts of (46) by P1, P2, Pa, P, and add: or the seconds by 1, 12, 13, 14 and add: either way we find .(55). P1Y1 +P2Y2 +P3Y3 + P11 = 0.................. (6) Multiply the firsts of (46) by 9,, ., ., 9, and add: multiply the seconds by P1, P2, P3, P, and add: and compare the results we find nΣp2 = nΣq2 = Σ12 (P ̧¶1 — P ̧¶‚)..............................................(56), where of the last member denotes a sum of such double terms as the sample without repetition of their equals, such as 21 (P1I, — P2l1). nalities proved between two components of one fundamental oscillation: and equality of their energies. (c) Let n2, n' denote the two values of - λ3 given in (32), Orthogoand let (54) and 1 nalities proved between different fundamental oscilla 41 =p', sin n't + q', cos n't, y,p', sin n't + q', cos n't, etc........(57) Case of equal periods. = 345 The case of n-n' is interesting. The equations Eq'q=0, Ep'p=0, Ep'q=0, Σq'p=0, when n differs however little from n', show (as we saw in a corresponding case in § 343 m) that equality of n to n' does not bring into the solution terms of the form Ct cos nt, and it must therefore come under § 343e. The condition to be fulfilled for the equality of the roots is seen from (32) and (33) to be 12 = 34, 13 = 42, and 14 = 23........ and to give ....(59): ...(60) for the common value of the roots. It is easy to verify that these relations reduce to zero each of the first minors of (42), as they must according to Routh's theorem (§ 343e), because each root, A, of (42) is a double root. According to the same theorem all the first, second and third minors of (47) must vanish for each root, because each root, n2, of (47) is a quadruple root: for this, as there are just four equations, it is necessary and sufficient that 11 = 22 = 33 = 44 and 120, 130, 140, 23 = 0, etc....(60'), which we see at once by (48) is the case when (59) are fulfilled. In fact, these relations immediately reduce (51) to 2 G (aa) = } (122 + 13a + 14°) (a,* + a ̧3 + a3 + a ̧3).................... (61). In this case one particular solution is readily seen from (52) and (46) to be Completed solution for case of equal periods. Hence the general solution, with four arbitraries p1, P2, P3, P ̧, is It is easy to verify that this satisfies the four differential equations (53). 345. Quite as we have dealt with (42), (45), (53), (54) in Two higher, § 345x, we may deal with (44) and the simple simultaneous equations for the solution of which they serve, which are dealt with of two and all the formulas which we meet in so doing are real when similarly w1, w, з, ш are all of one sign, and therefore 12', 13', etc., all by solution real. In the case of some of the 's negative and some positive similar quadratics, there is no difficulty in realizing the formulas, but the consideration of the simultaneous reduction of the two quadratics, ((12a+13 a ̧ +14a) } { (12 + (21α, +23 a +24α) + etc. (65), 2 to which we are led when we go back from the notation 12', etc. of (36), is not completely instructive in respect to stability, as was our previous explicit working out of the two roots of the determinantal equation in (37), (38), and (40). that gyro fluence be nant. 345xvill. The conditions to be fulfilled that the system may be provided dominated by gyrostatic influence are that the smaller value of static in-λ found from (31) and the greater found from (34) be re- fully domispectively very great in comparison with the greatest and very small in comparison with the smallest, of the four quantities ,,,, w1, w1 irrespectively of their signs. Supposing w, to be the greatest and the smallest, these conditions are easily proved to be fulfilled when, and only when, and 4 When these where >> denotes " very great in comparison with." Limits of smallest and second smallest of the four periods. will be diminished each in the inverse ratio, 1 to N. Again, let √1, √,, √±w, √w, be each diminished in the ratio M to 1; the two larger values of n will be sensibly unaltered; and the two smaller will be diminished in the ratio Ꮇ to 1. 345xix (a) each and Remark that When (66) is satisfied the two greater values of ʼn are < √/{(122 + 13a + 14° + 34° + 42o + 23')} 12.34 +13.42 + 14.23 > √( 122 + 13° + 14° + 34° + 422 + 233) } (68); and that when they are very unequal the greater is approximately equal to the former limit and the less to the latter. (b) When (67) is satisfied, and when the equilibrium is stable, the two smaller values of n are each √(12*w ̧w ̧+13°ŵ ̧w ̧+14aw‚w ̧+343w‚w ̧+42°œ‚w ̧+233w‚w) and (69), Limits of the next greatest and greatest of the four periods. Quadruply free cycloídal system with nondominant gyrostatic influences. √{(123w ̧w ̧+13°w ̧ ̧+14°œ ̧ ̧+34°π ̧ ̧+423 ̧ ̧+23°,~,)} and that when they are very unequal the greater of the two is approximately equal to the former limit, and the less to the latter. 345. Both (66) and (67) must be satisfied in order that the four periods may be found approximately by the solution of the two quadratics (31), (34). If (66) is satisfied but not (67), the biquadratic determinant still splits into two quadratics, of which one is approximately (31) but the other is not approximately (34). Similarly, if (67) is satisfied but not (66), the biquadratic splits into two quadratics of which one is approximately (34) but the other not approximately (31). 345xxi When neither (66) nor (67) is fulfilled there is not generally any splitting of the biquadratic into two rational quadratics; and the conditions of stability, the determination of the fundamental periods, and the working out of the complete solution depend essentially on the roots of a biquadratic equation. When w,,,,,, w, are all positive it is clear from the equation 4 |