t t Experiment VI.-Twisted for 121 minutes. 1 을 1 2 3 4 5 7 Scale divisions. 191 170 148 136 1264 1194 1081 ... 10 15 30 65 90 120 589 Scale divisions. 97 84 63 413 34 28 3} It should be mentioned that the operation of putting on the twist and of releasing each occupied about two seconds, and was performed half in the second before the epoch t = 0, and half in the second after or as nearly so as could be managed. The time was taken by ear from a clock beating seconds very distinctly. 3. The first point to be ascertained from these results is whether or not the principle of superposition, assumed by Boltzmann, holds for torsions of the magnitude here used. If the fibre be twisted for time T through angle X, then the torsion at time t after release will be X {v (T+t)-()} where y (t) = /0 (t) dt. If now T=tı + tz + tz + .. we may express the effect of one long twist in terms of several shorter twists by simply noticing that X{}()-(t+T)}=X[{()-(t+t)} + {}(t+t)-*(t+t+t); +{}(t+t+t)-(t+t+t+ts)}+, &c. Apply this to the preceding results, calculating each experment from its predecessor. Let at be the value of Yo (T+t) -ý (t), that is, the torsion at time t, when free, divided by the impressed twist measured in same unit; we obtain the following five tables of comparison. Results for T=2 compared with those from T=1. 1 2 3 4 5 7 10 20 40 Xt observed.... 041 023 018 2t calculated 029 016 Results for T=5 compared with those from T=2 and T=1. 10 24 observed.... 0.00328 262 212 182 164 136 110 * t calculated... 0.00323 233 181 156 136 108 193 t 15 22 58 151 Xt observed.... 087 072 036 010 2t calculated... 066 047 Results for T=10 compared with those from T=5. 1 / 를 1 2 3 4 7 10 469 398 339 300 236 197 t 15 25 45 120 170 Xt observed.... 125 092 067 036 031 xt calculated. 161 130 088 Results for T=20 compared with those from T=10. 1 2 3 4 5 7 10 80 100 Results for T=121 compared with those from T=20. 1 1 2 3 4 5 7 1070 950 880 830 780 730 10 15 30 65 90 120 589 24 observed... 497 433 325 212 174 144 18 Xt calculated.. 670 600 500 380 350 In examining these results it must be remembered that those for small values of T are much less accurate than when T is greater, for the quantity' observed is smaller but is sabject to the same absolute error; any irregularity in putting on or releasing from the stress will cause an error which is a material proportion of the observed deflection. For this reason it would be unsafe to base a conclusion on the experiments with T=1 and T=2. The three last tables agree in indicating a large deviation from the principle of superposition, the actual effect being less than the sum of the separate effects of the periods o stress into which the actual period may be broken up. Kohlrausch finds the same to be the case for india-rubber, either greater torsions or longer durations give less after-effects than would be expected from smaller torsions and shorter periods. 4. Assuming with Boltzmann that $(t)= we have at timet after termination of a twist lasting time T, &t=A{log (T+t) - log t}, the logarithms being taken to any base we please. The results were t Tut plotted on paper, ær being the ordinate and log the abscissa; if the law be true we should find the points all lying on a straight line : through the origin. For each value for T they do lie on straight lines very nearly for moderate values of t; but if T is not small these lines pass above the origin. When t becomes large the points drop below the straight line in a curve making towards the origin. This devia A tion appears to indicate the form (t)= a being less than, but ta near to, unity. If a=0.95 we have a fairly satisfactory formula. A ta In the following Table the observed and calculated values of me when T=121 are compared, A' being taken as 0.032. 5 7 t 10 15 30 65 90 120 589 42 To show the fact that A' decreases as T increases if a be assumed con stant, I add a comparison when T=20, it being then necessary to take A'=0.037. t 1 2 3 4 5 7 10 0:00580 470 398 358 327 276 234 0.00607 485 422 370 337 285 233 t 15 25 40 188 140 111 185 125 089 60 80 100 085 072 066 067 052 041 A better result would in this case be obtained by assuming a=0.92, or =0.93 in the former case with A'=0.021. Probably the best result would be given by taking A constant, and assuming that a increases with T. Taking the formula o(t)=4 these experiments give values of A ranging from 0·0017 to 0.0022. Boltzmann for a fibre, probably of & quite different composition, gives numbers froin which it follows that A=0.0036. 5. In my paper on “Residual Charge of the Leyden Jar” that subject is discussed in the same manner as Boltzmann discusses the after-effect of torsion on a fibre, and it is worth remarking that the results of my experiments can be roughly expressed by a formula in which $(t)= For glass No. 5 (soft crown) a=0-65, whilst for No. 7 (light flint) it is greater; but in the electrical experiment no sign of a definite deviation from the law of superposition was detected. IV. “Note in correction of an Error in the Rev. Dr. Haughton's Paper Notes on Physical Geology. No. V” (“Proc. Roy. In my paper read 20th June last, and published in the “ Journal of the Royal Society,” there is an error in p. 450 which I wish to correct. Referring to the geometrical proof of Mr. Darwin's theorem, I state that from cusp to cusp of the cycloidal wabble occupies 152 days; this is an error, as it should be 305 days, as can be shown geometri. cally. Let yx, y'a', be two successive positions of the line joining the axes of rotation and figure; produce them to meet at C, which will be the centre of curvature, because yæ and y'z', are normals to the cycloidal arc yy'; it is well known that yC, (radius of curvature) is double ya (chord of generating circle) or double y'x; therefore the angle yxy' is double the angle yCy'; but yxy' measures the angular velocity of the wabble, when æ is supposed at rest; therefore the angular velocity of yæ is only half that of the wabble, if the axis of figure were at rest. Hence in 305 days, yæ will turn through 180° only, and not 360°. This correction, when introduced into my calculation of Mr. Darwin's problem, p. 182, will double the result, and give 19,350 years to represent the 19,200 years, found by Mr. Darwin. I would wish to add, that Mr. Darwin, in a letter to myself, proposes to call the cycloidal wabble described by him, a “lopsided wabble,” as distinguished from the simple circular “wabble” described by me; the one being caused by continuous motion of the axis of figure, and the other caused by sudden displacement of that axis. V. “Measurements of Electrical Constants. No. II. On the Specific Inductive Capacities of Certain Dielectrics.” Part I. By J. E. H. GORDON, B.A. Camb. Communicated by Professor J. CLERK MAXWELL, F.R.S. Received October 21, 1878. (Abstract.) A paper of mine with the above title was communicated to the Royal Society by Professor J. Clerk Maxwell, F.R.S., on March 9th, 1878. It was read on March 28th, and an abstract of it appeared in the “Proceedings."* In the course of the summer it was pointed out to me that owing to a mistake in the formula of calculation all the results were wrong. I, therefore, requested permission to withdraw my paper, in order to recalculate the results. The new values of K arrived at led me to make some determinations of refractive indices and to re-write the theoretical deductions at the close of the paper. I now beg through Professor Maxwell to present the paper in an amended form, in the hope that it may be found not entirely unworthy of the attention of the Royal Society. As it would be impossible within the limits of an abstract to give any intelligible account of the new method of experiment (due to Professor Maxwell), which has been employed, I will merely give the table of results, reserving all discussion and explanation until the publication of my paper in full. I may, however, state that the method is a zero method, that the electrified metal plates never touch the dielectrics, and that the electrification, which is produced by an induction coil, has an electromo. tive force equal to that of about 2,050 chloride of silver cells, and is reversed some 12,000 times per second. * Ante, vol. xxvii, p. 270. |