No. 1. No. 2. No. 3. No. 4. bifilarly slung in four ways. Four freedoms, reducible to three if desired by a third thread in each case, diagonal in the first and second, lateral in the third and fourth, the freedom thus annulled being in each case stable and independent of the rotation of the fly-wheel. Three modes essentially involved in the gyrostatic system in each case, two inclinational and one azimuthal. No. 1.-Azimuthally stable without rotation; with rotation all three modes stable. No. 2.-Azimuthally stable, one inclinational mode unstable the other stable without rotation; with rotation two unstable, one stable. No. 3.-The azimuthal mode unstable, two inclinational modes stable the other unstable, without rotation; with rotation one azimuthal mode and one inclinational mode unstable, and one inclinational mode stable. No. 4.-Azimuthally and one inclinational mode unstable, one inclinational mode stable, without rotation; with rotation all three stable. system with 345. Take for another example a system having three Gyrostatic freedoms (that is to say, three independent co-ordinates three free41, 42, 43), (16) become - 21. (27), where 91, 92, 9, denote the values of the three pairs of equals 3 The doms. Gyrostatic system with three freedoms: reduced to a mere rotating system. 2 where w = √(g,2+g+g), and the force-components parallel to the fresh axes are denoted by X, Y, Z (instead of dV dV dx dv because the present transformation is clearly indz dependent of the assumption we have been making latterly that the positional forces are conservative). These (28) are simply the equations [§ 319, Ex. (E)] of the motion of a particle relatively to co-ordinates revolving with angular velocity round the axis OZ, if we suppose X, Y, Z to include the components of the centrifugal force due to this rotation. Hence the influence of the gyroscopic terms however originating in any system with three freedoms (and therefore also in any system with only two freedoms) may be represented by the motion of a material particle supported by massless springs attached to a rigid body revolving uniformly round a fixed axis. It is an interesting and instructive exercise to imagine or to actually construct mechanical arrangements for the motion of a material particle to illustrate the experiments described in § 345*. 345xii Consider next the case of a system with four freedoms. The equations are Ÿ‚ + 124 ̧ + 13Ỷ ̧+ 14Ÿ‚ + @ ̧¥1 = 0 Denoting by D the determinant we have, by (18), D= (λ3+ w,) (λ3 + w2) (λ2 + w ̧) (λ3 + @1) I {34°(λ2+@,)(λ2+w ̧)+122(λ3+w3)(\3+w ̧)+422(\3+w,)(\2+wg) + 13°(\'+w ̧)(\'+w ̧)+23′(x2+w,)(x2+w) + 14′(x2+w,)(X°+w ̧)} +λ* (12 342+ 13 42 + 14 23)9 (29). ..(30). λ®+ (12° +13+14+23+42+34) λ+ (12 34+ 13 42+ 14 23)λ*. This equated to zero and viewed as an equation for X has two roots each equal to 0, and two others given by the residual Quadruply quadratic free gyro static system without λ*+ (1 22+ 1 33+ 142+23° +24a +34°) λ2 + (12 34+13 42 +14 23)=0...(31). force. Now remarking that the solution of x2+pz+q2=0 may be written · z = } { p ± √ (p + 2q) (p − 2q)} = } { √(p + 2q) ± √(p − 2q)}', we have from (31) where and - X2 = 3 ( 12′+ 13′+ 14′+ 23′+ 24*+ 34° ± √3) )})..............(32), = (r±8)3 r = √{( 12 + 34)2 + (1 3 + 42)2 + ( 1 4 +23)2} 8 = √ √ {( 1 2 − 34)2 + (1 3 − 42)2 + (14−23)"} } ...... (33). case of fail static pre As 12, 34, 13, etc. are essentially real, r and s are real, and (unless 12 43 + 13 42 + 14 23 = 0, when one of the values of A is Excepted zero, a case which must be considered specially, but is excluded ing gyrofor the present,) they are unequal. Hence the two values of dominance. -λ given by (32) are real and positive. Hence two of the four freedoms are stable. The other two (corresponding to -λ2=0) are neutral. 345xi Now suppose w1, w, W w to be not zero, but each Quadruply free cycloidvery small. The determinantal equation will be a biquadratic al system, in A, of which two roots (the two which vanish when w,, etc. cally domivanish) are approximately equal to the roots of the quadratic (12 34 + 13 42 + 14 23)3λ* + (123π ̧μ ̧+133a̸ ̧ ̧+143w ̧ ̧ +23°w,w ̧ +243 ̧ ̧ + 34° ̧ ̧)λ2 + w ̧ ̧ ̧0...(34), 2 = and the other two roots are approximately equal to those of the To solve equation (34), first write it thus : 1 : (~+)*+ (12"+13′′+14"+23′′+24′′+ 34") « +(12′34′+13′42′+14′23′′)" = 0 etc.......(36). VOL. I. 26 Quadruply free cycloidal system, gyrostatically dominated. Thus, taken as a quadratic for A-2, it has the same form as (31) for A, and so, as before in (32) and (33), we find Four irro tational stabilities confirmed, four irrotational instabilities rendered stable, by gyrostatic links. (37), ..(38). Now if w,,,,,,, be all four positive or all four negative, 12', 34', 13', etc. are all real, and therefore both the values of 1 given by (37) are real and positive (the excluded case referred to at the end of § 345, which makes 12'34' + 13′42′ + 14′23′ = 0, 1 4 and therefore the smaller value of -= 0, being still excluded). Hence the corresponding freedoms are stable. But it is not necessary for stability that a,, w ̧, w1, w1 be all four of one sign: it is necessary that their product be positive: since if it were negative the values of X given by (34) would both be real, but one only negative and the other positive. Suppose two of them,,,, for example, be negative, and the other two, positive: this makes ~,~,, w11, w ̧ ̧, and w ̧ ̧ negative, and therefore 13', 14', 23', and 24' imaginary. Instead of four of the six equations (36), put therefore Thus 13" etc. are real, and 13' 13"-1 etc., and (38) become Combined dynamic and gyrostatic sta If these inequalities are reversed, the stabilities due to w1, w, and 34′ are undone by the gyrostatic connexions 13′′, 42′′, 14′′ and 23". statically acted. 345. Going back to (29) we see that for the particular bility gyrosolution 1 = a,e^t, 1⁄4 ̧= a ̧1t, etc., given by the first pair of roots counterof (32), they become approximately solution. being in fact the linear algebraic equations for the solution in Completed of the simple simultaneous differential equations the form for either particular approximate solution of (29) corresponding to (37), we find from (29) approximately Remark that in (42) the coefficients of the first terms are imaginary and those of all the others real. Hence the ratios a/a,, a,/a,, etc., are imaginary. To realize the equations put x=n√− 1, a1 =p1 + q1 √ − 1, a2 = p2 + q1 √-1, etc....(45), and let P1, I, P, etc. be real; we find, as equivalent to (42), 13 14 S-ng np +1292 +1393 +144 0 nq,+2IP, + 23 Pa +24 P1 +21 9+233 + 24 etc. = Eliminating 1,,,, etc. from the seconds by the firsts of these pairs, we find Realization of completed solution. and by eliminating P,, P., etc. similarly we find similar equations |