system with conserva tive posi and unre motional configurations with such velocities that when to it shall be Cycloidal at rest in its zero configuration: this we see by taking, as a particular solution, the terms of (9) § 345iv below, for which m is tional forces negative. But this equilibrium is essentially unstable, unless V stricted is positive for all real values of the co-ordinates. To prove this forces. imagine the system placed in any configuration in which V is negative, and left there either at rest or with any motion of kinetic energy less than or at the most equal to - V: thus E. will be negative or zero; T+ V will therefore have increasing negative value as time advances; therefore V must always remain negative; and therefore the system can never reach its zero configuration. It is clear that V and T must each on the whole increase though there may be fluctuations, of T diminishing for a time, during which must also diminish so as to make the excess (V) – T increase at the rate equal to per unit of time according to formula (2). 345. To illustrate the circumstances of the several cases let λ = m + n √−1 be a root of the determinantal equation, m and n being both real. The corresponding realized solution of the dynamical problem is = 2 2 41 = r1e cos (nt-e1), 4, reTM cos (nt - e), etc..........(5), where the differences of epochs e, e1, e̟-è̟,, etc. and the ratios r/r1, etc., in all 2-2 numerics *, are determined by the 2i simultaneous linear equations (3) of § 343 harmonized by taking for λ=m+n√−1, and again λ=m-n√-1. Using these expressions for 4,,,, etc. in the expressions for V, Q, T, we find, V=2 (C + A cos 2nt + B sin 2nt) 66 The term numeric has been recently introduced by Professor James Thomson to denote a number, or a proper fraction, or an improper fraction, or an incommensurable ratio (such as T or e). It must also to be useful in mathematical analysis include imaginary expressions such as m+n √-1, where m and n are real numerics. " Numeric" may be regarded as an abbreviation for "numerical expression." It lets us avoid the intolerable verbiage of integer or proper or improper fraction which mathematical writers hitherto are so often compelled to use; and is more appropriate for mere number or ratio than the designation "quantity," which rather implies quantity of something than the mere numerical expression by which quantities of any measurable things are reckoned in terms of the unit of quantity. Cycloidal system with conserva tive posi tional forces and unrestricted motional forces. where C, A, B, C, A', B', C", A", B", are determinate constants: and in order that Q and T may be positive we have C' > + √(A'2 + B'), and C">+ √(A"2 + B'3)...........(7). Substituting these in (2), and equating coefficients of corresponding terms, we find 2m (C + C") = − C' Real part of every root proved negative when V positive for all real co-ordinates; positive for some roots when has negative values; but always negative for some roots. The first of these shows that C + C" and m must be of contrary signs. Hence if V be essentially positive [which requires that C be greater than + √(A3 +B2)], every value of m must be negative. 345iv. If V have negative values for some or all real values of the co-ordinates, m must clearly be positive for some roots, but there must still, and always, be roots for which m is negative. To prove this last clause let us instead of (5) take sums of particular solutions corresponding to different roots x = m ±n √-1, X' = m'n' √-1, etc., m and n denoting real numerics. Thus we have Suppose now m, m', etc. to be all positive: then for t∞, we should have =0, 2=0, 41 =0,4-0, etc., and therefore V=0, T=0. Hence, for finite values of t, T would in virtue of (4) be less than (which in this case is essentially positive): but we may place the system in any configuration and project it with any velocity we please, and therefore the amount of kinetic energy we may give it is unlimited. Hence, if (9) be the complete solution, it must include some negative value or values of m, and therefore of all the roots λ, X', etc. there must be some of which the real part is negative. This conclusion is also obvious on purely algebraic grounds, because the coefficient of A2-1 in the determinant is obviously 11+22+33 +..., which is essentially positive when Q is positive for all real values of the co-ordinates. 345. It is an important subject for investigation, interesting both in mere Algebra and in Dynamics, to find how many roots there are with m positive, or how many with m negative in any particular case or class of cases; also to find under what con latory sub stable equi falling away stable. to stable ditions n disappears [or the motion non-oscillatory (compare Non-oscil§ 341)]. We hope to return to it in our second volume, and sidence to should be very glad to find it taken up and worked out fully by librium, or mathematicians in the mean time. At present it is obvious that from unif V be negative for all real values of 1, V, etc., the motion must Oscillatory be non-oscillatory for every mode (or every value of A must be subsidence real) if Q be but large enough: but as we shall see immediately equili with Q not too large, n may appear in some or in all the roots, falling away even though be negative for all real co-ordinates, when there stable. are forces of the gyroscopic class [§ 319, Examp. (G) above When the motional forces are wholly of § 345 below). viscous class it is easily seen that n can only appear if positive for some or all real values of the co-ordinates: n disappear if V is negative for all real values of the co-ordinates (again compare § 341). brium, or from un Falling and away from the stable equiis essentially wholly un must librium is non-oscillatory if motional forces wholly viscous. Dissipative 345. A chief part of the substance of SS 345"... 345" above may be expressed shortly without symbols thus:-When there is any dissipativity the equilibrium in the zero position is stable or unstable according as the same system with no motional stability of forces, but with the same positional forces, is stable or unstable. system. The gyroscopic forces which we now proceed to consider may convert instability into stability, as in the gyrostat § 345' below, when there is no dissipativity:—but when there is any dissipativity gyroscopic forces may convert rapid falling away from an unstable configuration into falling by (as it were) exceedingly gradual spirals, but they cannot convert instability into stability if there be any dissipativity. The theorem of Dissipativity [§ 345', (2) and (3)] suggests the following notation, | (12 + 21) = [12] or [21], (13 + 31) = [13] or [31], etc. and (12 −21) = 12] or − 21], } (13 – 31) = 13] or – 31], etc.) (10), so that the symbols [12], [21], [13], etc., and 12], 21], 13], etc. [12]=[21], [13]=[31], [23]=[32], etc.) Thus (3) of § 3451 becomes (11). Q =11+2 [I2], +224 +2[3]. +etc.......(12), 3 and going back to (1), with (10) and (12) we have Various origins of gyroscopic terms. Equation of energy. In these equations the terms 12] Ý„, 21]Ý,, 13] Ý„, 31]†,, etc. represent what we may call gyroscopic forces, because, as we have seen in § 319, Ex. G, they occur when fly-wheels each given in a state of rapid rotation form part of the system by being mounted on frictionless bearings connected through framework with other parts of the system; and because, as we have seen in § 319, Ex. F, they occur when the motion considered is motion of the given system relatively to a rigid body revolving with a constrainedly constant angular velocity round a fixed axis This last reason is especially interesting on account of Laplace's dynamical theory of the tides at the foundation of which it lies, and in which it is answerable for some of the most curious and instructive results, such as the beautiful vortex problem presented by what Laplace calls "Oscillations of the First Species*." 345. The gyrostatic terms disappear from the equation of energy as we see by § 345', (2) and (3), and as we saw previously by § 319, Example G (19), and in § 319, Ex. F' (ƒ). Comparing § 319 (ƒ) and (g), we see that in the case of motion relatively to a body revolving uniformly round a fixed axis it is not the equation of total absolute energy but the equation of energy of the relative motion that the gyroscopic terms disappear from, as (f) of § 319; and (2) and (3) of § 345 when the subject of their application is to such relative motion. * The integrated equation for this species of tidal motions, in an ideal ocean equally deep over the whole solid rotating spheroid, is given in a form ready for numerical computation in "Note on the Oscillations of the First Species' in Laplace's Theory of the Tides" (W. Thomson), Phil. Mag. Oct. 1875. conservative 345vili. To discover something of the character of the gyro- Gyrostatic scopic influence on the motion of a system, suppose there to be system: no resistances (or viscous influences), that is to say let the dissipativity, Q, be zero. The determinantal equation (4) becomes +11, (12)2+12]λ +12,... (11)λ (21)λ+21]λ + 21, (22)λ* + 22,... = 0......(14). Now by the relations (12)=(21), etc., 12 = 21, etc., and 12]=-21], we see that if λ be changed into the determinant becomes altered merely by interchange of terms between columns and rows, and hence the value of the determinant remains unchanged. Hence the first member of (14) cannot contain odd powers of λ, and therefore its roots must be in pairs of oppositely signed equals. The condition for stability of equilibrium in the zero configuration is therefore that the roots λ2 of the determinantal equation be each real and negative. tion of its 345. The equations are simplified by transforming the co- simplificaordinates (§ 337) so as to reduce T to a sum of squares with equations. positive coefficients and to a sum of squares with positive or negative coefficients as the case may be, or which is the same thing to adopt for co-ordinates those displacements which would correspond to "fundamental modes" (§ 338), if the positional forces were as they are and there were no motional forces. Suppose farther the unit values of the co-ordinates to be so chosen that the coefficients of the squares of the velocities in 27 shall be each unity; and let us put w1, w2, wz9 etc. instead of the coefficients 11, 22, 33, etc., remaining in 2V. Thus we have T = (412 +422 + etc.), and V = (@‚,4,2 + @,4,2 + etc.)..... (15). If now we omit the half brackets] as no longer needed to avoid ambiguity, and understand that 12=- 21, 13=-31, 23=-32, etc., the equations of motion are |