3452 Quite as we have dealt with (42), (45), (53), (54) in Two higher, $ 345*, we may deal with (44) and the simple simultaneous equations for the solution of which they serve, which are of two and all the formulas which we meet in so doing are real when similarly T0,5 W,, wg, w, are all of one sign, and therefore 12', 13', etc., all by solution real. In the case of some of the w's negative and some positive similar there is no difficulty in realizing the formulas, but the con quadratics, sideration of the simultaneous reduction of the two quadratics, Ž (12 a,+ 13 a,+ 14 a.) (210, +23 a3 + 24 a) + etc. w (65), and 1 (w,a,' + w,a,' + w,a,' + w,a,') 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). + W 2 nant, 345 xviii. The conditions to be fulfilled that the system may be provided dominated by gyrostatic influence are that the smaller value of static in. cdo found from (31) and the greater found from (34) be re- fully domi. spectively very great in comparison with the greatest and very emall in comparison with the smallest, of the four quantities w, wg, w, w, irrespectively of their signs. Supposing w, to be the greatest and w, the smallest, these conditions are easily proved to be fulfilled when, and only when, (12 . 34 + 13 42 + 14. 23) >>= 0, ... .......(66), ->>#w, (67), 12’ww.+13'w.w,+ 14'w,w,+ 34%w, w,+42'w, w,+23*w, w, where >> denotes “very great in comparison with.” When these conditions are fulfilled, let 12, 13, 23, etc., be each increased in the ratio of N to 1. The two greater values of n (or 1 -1) will be increased in the same ratio, N to l; and the two smaller will be diminished each in the inverse ratio, 1 to N. Again, let 1+w,, 7+2, 1+ wg, +w, be each diminished in the ratio to l; the two larger values of n will be sensibly unaltered; and the two smaller will be diminished in the ratio V to 1. Limits of suallest and second smallest of the four periods. } 345. Remark that (a) When (66) is satisfied the two greater values of n are each </{(12° + 13' + 14* + 34' + 42° +23°)} (68); and J(12° + 13' + 14' + 34' + 42' + 23) and that when they are very unequal the greater is approximately equal to the former limit and the less to the latter. (6) When (67) is satisfied, and when the equilibrium is stable, the two smaller values of n are each \(12ľa,, + 13*2.2,+ 148w,w,+34'w, 2, +42*w,a,+23*,,) 12 . 34 + 13.42 + 14.23 and (69), Jlco, www.) _{(12'ww.+13*w,w,+14'WW+34'w,w, +42*w,ws+238w,w)} 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. Limits of the next greatest and greatest of the four periods. 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), (84). 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). Quadruply free cycloi. dal system with nondominant gyrostatic influences. 345vi 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,, Wg, wg, w, are all positive it is clear from the equation of energy (345", (4), with Q=0] that the motion is stable what- Quadruply free cycluiever be the values of the gyrostatic coefficients 12, 34, 13, etc. dal spstem and therefore in this case each of the four roots lê of the biquad- dominant ratic is real and negative, a proposition included in the general influences. theorem of $ 3456xvi below. To illustrate the interesting questions which occur when the a's are not all positive put 12 = 129, 34 = 349, 13 = 139, etc.... ..(70), where 12, 34, 13, etc. denote any numerics whatever subject only to the condition that they do not make zero of 12 . 34 + 13.42 + 14.23 When w,, WE, Wy, a are all negative each root lo of the biquadratic is as we have seen in $ 345dii real and negative when the gyrostatic influences dominate. It becomes an interesting question to be answered by treatment of the biquadratic, how small may g be to keep all the roots dreal and negative, and how large may g be to render them other than real and positive as they are when g=0? Similar questions occur in connexion with the case of two of the w's negative and two positive, when the gyrostatic influences are so proportioned as to fulfil 345tii (41), so that when g is infinitely great there is complete gyrostatic stability, though when g=0 there are two instabilities and two stabilities. number of 345d. Returning now to 345% and 345, 345vili, and 345", Gyrostatio for a gyrostatic system with any number of freedoms, we see by with any 345" that the roots lp of the determinantal equation (14) or (17) freedoms. are necessarily real and negative when w,, W,, , W., etc. are all positive. This conclusion is founded on the reasoning of $ 345' regarding the equation of energy (4) applied to the case Q=0, for which it becomes T + 1 = E., or the same as for the case of no motional forces. It is easy of course to eliminate dynamical considerations from the reasoning and to give a purely algebraic proof that the roots lp of the determinantal equation (14) of 345vili are necessarily real and negative, provided both of the two quadratic functions (11) a,' +2 (12) a,a, + etc., and 11a,' +2 12a,a,+ etc. are positive for all real values of a,,a,, etc. But the equations (14) of 8 343 (k), which we obtained and used in the course of the corresponding demonstration for the case of no motional forces, do not hold in our present case of gyrostatic motional forces. Still for this present case we have the con Case of equal roots with slabuity. clusion of $ 313 (m) that equality among the roots falls essentially under the case of $ 343 (e) above. For we know from the consideration of energy, as in § 345*, that no particular solution can be of the form telt or t sin ot, when the potential energy is positive for all displacements: yet (though there cannot be equal roots for the gyrostatic system of two freedoms (S 345") as we see from the solution (25) of the determinantal equation for this case) there obviously may be equality of roots * in a quadruply free gyrostatic system, or in one with more than four freedoms. Hence, if both the quadratic functions have the same sign for all real values of a,, ac, etc., all the first minors Application of Ruth's theorem, Examples of this may be invented ad libitum by commencing with pairs of equations such as (23) and altering the variables by (generalized) orthogonal transformations. For one very simple example put š=w and take (23) as one pair of equations of motion, and as a second pair take t'n'W'=0, 1 - ชั+ 2 = 0. The second of (23) and the first of these multiplied respectively by cosa and sin a, and again by sin a and cos a, and added and subtracted, give 7z-cos aš +y sin an' + @4o=0, and Hs+sin aš + y cos ar' + 43=0, where Yo=sin a + n cos a, and 43=' cos a -msin a. Eliminating g' and n by these last equations, from the first and fourth of the equations of motion, and for symmetry putting y, instead of g, and 4: instead of r', and for simplicity putting y cos a=g, and y sin a=h, and collecting the equations of motion in order, we have the following, +gV- 4x + 4y=0, 3.- 4+hy+wy,=0, 7s+h, +94. + wys=0, -hy, -94, + wyn=0, for the equations of motion of a quadruply free gyrostatic system having two equalities among its four fundamental periods. The two different periods are the two values of the expression 27/{V(+ 7) + (19+ the +w). When these two values are unequal the equalities among the roots do not give rise to terms of the form tedt or tcos ot in the solution. But if W= -(+92 + Phạ), which makes these two values equal, and therefore all four roots equal, terms of the form t cos ot do appear in the solution, and the equili. brium is unstable in the transitional case though it is stable if @ be less than 19? + fl? by ever so small a difference. of Routli's of the determinantal equation (14), $ 345vil, must vanish for each Applicntion double, triple, or multiple root of the equation, if it has any theorem, such roots. It will be interesting to find a purely algebraic proof of this theorem, and we leave it as an exercise to the student; remarking only that, when the quadratic functions have contrary signs for some real values of a,, Q,, etc., there may be equality among the roots without the evanescence of all the first minors; or, in Equal roots dynamical language, there may be terms of the form tent, or bility in transitional i sin ot, in the solution expressing the motion of a gyrostatic system, in transitional cases between stability and instability. bility and It is easy to invent examples of such cases, taking for instance instability. the quadruply free gyrostatic system, whether gyrostatically dominated as in 345xit, but in this case with some of the four quantities negative, and some positive; or, as in § 345», not gyrostatically dominated, with either some or all of the quantities ,, W,, ..., w, negative. All this we recommend to the student as interesting and instructive exercise. with insta cases bas tween sta of gyro: static do. 345. When all the quantities w,, Wg, ..., W; are of the Conditions same sign it is easy to find the conditions that must be fulfilled in order that the system may be gyrostatically dominated. For mination. if Pie Pg, ..., Pn are the roots of the equation 0,7 +0,3"-) + ... +C-,+c, = 0, Hence if-P1, P2, ... -Pn be each positive, c/nc, is their arithmetic mean, and nc, / Cn-, is their harmonic mean. Hence c/ne, is greater than nc, / 0,-1, and the greatest of -p,, - PR,..., - po is greater than c, / nco, and the least of them is less than ne, / Cm Take now the two following equations: x'+/- E 12° +2"-* £ (£12.34)' +1'- (212.34. 56)* + etc. = 0 .........(71), 6 ) +4)***£12"+ 2"+5)** £¢£12' 347+()*3(512.34.56%+etc.= 0 (72), |