mercial" copper and that which is sold by chemical supply houses under the label "chemically pure.' From the tables it will be seen that the plates are practically clear except for the impurity lines, which are very weak, many of them not showing on a silver print. In any case a table of the impurity lines and "ghosts" might accompany each map. A few years hence, when the spectra of the metals are more completely measured, such a table will be easily made. North-Western University, Evanston, Illinois, U.S.A. By XLIV. On the Vibrations of a Loaded Spiral Spring. IT T has been pointed out by Profs. Ayrton and Perry † that, by comparing the axial elongation and the twisting produced in a spiral spring of finite angle by the action of an axial force, we can deduce the ratio of the torsional and flexural rigidities of the wire or strip of which the spring is made, and hence obtain the ratio of the rigidity to the Young's modulus of its material. This method is very interesting and instructive; but as it is not easy to produce springs of convenient and yet sufficiently uniform angles, nor to determine accurately a small axial elongation, it seemed to me that it might be worth while to modify it by attaching a mass to the spring and observing the periods of the vibrations which it executes when displaced. In this case it will be found convenient to use a spring of an angle so small that its square may be neglected. Apart from their use in comparing moduli of elasticity, the vibrations of such a system present some rather interesting features, of which a detailed consideration may not be out of place. If we have a spiral spring made of a length l of wire, and wound on a cylinder of radius r, so that the distance between the ends of the spring is a, and if is the angle between the planes through the axis of the spiral and the two ends of the wire, the force and couple required to produce a deformation from the equilibrium state (o, Po) to the state *Communicated by the Author. where A is the torsional and B the flexural rigidity of the If the spring is hung up in a vertical position with its Fx,q=Mg, Cx, ¢=0. Now let the spring be displaced to the configuration (x+dx, where Mk2 is the moment of inertia of M. about the axis of If the angles of the unstretched and of the stretched spring are both so small that we can neglect it follows from the * Thomson and Tait's 'Natural Philosophy,' vol. i. part ii. p. 141. where a and c are necessarily positive, and b is small. The most general form of solution is Sx=A1 sin pt + A, cos pt + B1 sin qt + B2 cos qt, a Sk$=12=a(A ̧ sin pt +A, cos pt) +2=a (B, singt + B2cos qt), b b where p2 and q2 are the roots of the quadratic (x—a) (x—c)=b2, We will suppose p2 to be the greater. The solution may be put into the form out the motion, and in the latter 2-8x=8kp. b δα=δίφ. It is easily seen that p2-a is positive and q2-a negative; therefore we conclude that, if b is a positive quantity, the shorter period of vibration corresponds to a screwing motion similar to the screw of the spring, and the longer to a screwing motion opposite to the screw of the spring, while if 6 is negative the reverse is the case. It is also clear that if (c-a) is large is is large small}, and b small compared with b and qa is large; thus shorter} period cor case the vibrations of respond to da =0, and those of longer period to 84 =0, approximately. c~a shorter If, however, is finite and equal to 22, we have, when b the system is vibrating in one of its normal modes, either throughout the motion, the periods corresponding to these modes being nearly equal. In this case, if the system receives a displacement not represented by either of these equations, the subsequent motion will be compounded of two vibrations, one of which slowly gains upon the other, and will thus exhibit phenomena of intermittence. For example, if the displacement (dx=X, d4=0) be given, this may be resolved into and therefore, when the vibrations of one normal mode have gained half a period on those of the other, the half-amplitude of the x-vibration will have decreased from X to X λ will have appeared, while when another half-period is gained the initial conditions will be restored. Thus, while at first the system moves simply with an x-vibration, this gradually diminishes to a minimum value, and at the same time a -vibration is gradually set up and grows to a maximum; the latter vibration then decreases and finally vanishes, while the former increases until it reaches its initial value, and then the phe nomena recur. It is easy to see that a similar intermittence will be exhibited if the system is started with a p-vibration only. The above results may readily be verified experimentally by employing as the mass M a body of adjustable moment of inertia. The most interesting case is that in which k is adjusted so that λ is rendered very small, when the energy is seen to be transferred with almost perfect completeness from a-vibrations to -vibrations and back again. The condition for the vanishing of A is of course A62 B = ; 13 lk2 or, since to our order of approximation l=rp, B It is also of interest to produce, by means of a series of suitably timed small impulses, the normal modes of vibration of such a system and to demonstrate the permanence of each. If, however, the object in view is the determination of elastic constants, it is convenient to arrange that (c-a) shall be large compared with b. In this case, as we have seen, pure a-vibrations and pure -vibrations are practically the two normal modes, and the periodic times of the former and the latter are given by 2π If the mass m of the spring itself cannot be neglected, we can allow for it, if small compared with M, by taking M+3m as the vibrating mass, and Mk2+ m2 as its moment of inertia*. Let us consider the case of a spring made of circular wire of radius p. If we may assume the material to be homogeneous and isotropic, an assumption which is undoubtedly a weak point of all methods of determining the elastic constants of a material by experiments on wires, we have where E and n are respectively the Young's modulus and the rigidity of the material. From the above we obtain an equation involving only quantities easy of measurement, and hence Poisson's ratio, which is equal to ≥( determined. * Lord Rayleigh's 'Theory of Sound,' § 156. E n -2), is |