ELASTICITY 803 a single vibration through one range was greater the greater the velocity (within the limits of the experiments) ; but the difference between the losses at low and high speeds was much less than it would have been had the resistance been, as Stokes has proved it to be, in fluid friction, approximately as the rapidity of the change of shape. The irregularities in the results of the experiments which up to this time I have made seem to prove that much smaller vibrations (producing less absolute amounts of distortion in the parts of the wires most stressed) must be observed before any simple law of relation between molecular friction and velocity can be discovered. (b) " When the weight was increased, the viscosity was always at first much increased ; but then day after day it gradually diminished and became as small in amount as it had been with the lighter weight. It has not yet been practicable to continue the experiments long enough in any case to find the limit to this variation. (c) " The vibration subsided in aluminium wires much more rapidly from amplitude 20 to amplitude 10, when the initial amplitude was 40, than when it was 20. Thus, with a certain aluminium wire, and vibrator No. 1 (time of vibration one way 1757 second), the number of vibrations counted were in three trials Vibrations. Subsidence from 40 initial amplitude to 20 56 64 64 And from 20 (in course of the same experiments) ) Ol , QQ QA to 10 Jb yb y The same wire and the same vibrator showed Subsidence from 20 initial amplitude to 10 ) ,, (average of four trials) [ 112 ^rations. Agnin, the same wire, with vibrator No. 2 1 (time of vibration one way 1 236), showed in two trials Vibrations. Subsidence from 40 initial amplitude to 20 54 52 And continued from 20 to 10 90 90 Again, same wire and vibrator, From initial amplitude 20 to 10 . . 103 (mean of eight trials). This remarkable result suggested the question (d). (d) "In a wire which was kept vibrating nearly all day, from day to day, after several days very much more mole cular friction was found than in another kept quiescent ex cept during each experiment. Thus two equal and similar pieces of copper wire were put up about the 26th of April, hanging with equal and similar lead weights, the upper and lower ends of the two wires being similarly fixed by solder ing. No. 2 was more frequently vibrated than No. 1 for a few clays at first, but no comparison of viscosities was made till May 15. Then No. 1 subsided from 20 initial range to 10 in 97 vibrations. No. 2 gave the same subsidence in 77 vibrations. During the greater part of May 16 and 17, No. 2 was kept vibrating and No. 1 quiescent, and late on May 17 experi ments with the following results were made : Time per Vibration. No. 1 subsided from 20 to 10 after 99 vibrations in 237 sees., 2 4 2 subsided from 20 to 10 a 7O 98 OU 235 A 2-4 ter 58 vibrat ons in 142 2-45 60 147 2 45 57 139 2-45 60 147 2-45" No. [Addition, May 27, after the reading of the paper.] No. 1 has been kept at rest from May 17, while No. 2 has been kept oscillating more or less every day till yesterday, May 26, when both were oscillated, with the following results : Time per Vibration. ko. 1 subsided from 20 to 10 after 100 vibrations in 242 sees., 2 42 ^o.2 ,, ,, 44 or 45 vibrations 2 495 35. The investigation was continued with much smaller degrees of maximum angular distortion, to discover, if 1 Of same weight as No. 1, but different moment of inertia. possible, the law of the molecular friction, the existence of which was demonstrated by these experiments. Two ques tions immediately occurred : What is the law of sub sidence of range in any single series of oscillations, the vibrator being undisturbed by external force 1 ? and (ques tion (a) of 33 above) what is the relation between the law of subsidence in two sets of oscillations having different periods, with the same elastic body in the same circum stances of elastic force, as for instance the same or similar metallic wires with equal weights hung upon them, per forming torsional oscillations in different times on account of the moments of inertia of the suspended masses being different ] 36. So far as the irregularities depending on previous conditions of the elastic substance allowed any simple law to be indicated, the experimental answer to the first question for degrees of angular distortion much smaller than the palpable limits of elasticity was the COMPOUND INTEREST LAW, that is to say, The diminutions of range per equal intervals of time or per equal numbers of oscilla tions bore a constant proportion to the diminishing range ; or, The differences of the logarithms of the ranges were pro portional to the intervals of time. The only approach to an answer to the second question yet obtained is that the proportionate losses of amplitude in the different cases are not such as they would be if the molecular resistance were simply proportional to the velocity of change of shape in the different cases. If the molecular friction followed this simple law, the proportionate diminu tions of range per period would be directly as the square roots of the periods, or per equal intervals of time they would be inversely as the square roots of the periods. Instead of the proportion being so, the loss wac greater with the longer periods than that calculated according to the law of square roots from its amount in the shorter periods. It was in fact as it would be if the result were wholly or partially due to imperfect elasticity, or " elastische Nach-wirkung " elastic after-working as the Germans call it (compare section 6 above). To form a rough idea of the results, irrespectively of the ultimate molecular theory (which is to be looked for in the proper extension of Maxwell s kinetic theory of viscosity of gases), consider a perfectly elastic vesicular solid, whether like a sponge with communications between the vesicles, or with each vesicle separately inclosed in elastic solid : imagine its pores and interstices filled up with a viscous fluid, such as oil. Static experiments on such a solid will show perfect elasticity of bulk and shape ; kinetic experiments will show losses of energy such as are really shown by vibrators of india-rubber, jelly, glass, metals, or other elastic homogeneous solids, but more regular, and following more closely the compound interest law for single series and the law of relation to square roots of periods stated above for sets of oscillations in different periods. In short, according to Stokes s law of viscosity of fluids, our supposed vesicular vibrator would follow the law of subsidence of a simple vibrator experiencing a resistance simply proportional to the velocity of its motion, while no such simple law is applicable to the effects of the internal molecular resistance in a vibrating elastic solid. 37. Ilooke s Law. A law expressed by Hooke with Latin terseness in the words Ut tensio sic vis is the foundation of the mathematical theory of the elasticity of hard solids. By tensio here is meant not force (as is generally meant by the English word tension), but an elongation produced by force. In English, then, Hooke s law is that elongation (understood of an elastic solid) is proportional to the force producing it. It is, of course, to be extended continuously from elongation to coutraction
in respect to the effect, and from pull to push in. respect