observed after-effect was not due to the fibre twisting within the clamps and then sticking. The difficulty was easily avoided by employing two mirrors, each cemented at a single point to the glass fibre itself, one just below the upper clamp, the other just above the lower clamp. The upper mirror merely served by means of a subsidiary lamp and scale to bring back the part of the fibre to which it was attached to its initial position. The motion of the lower clamp was damped by attaching to it a vane dipping into a vessel of oil. The temperature of the room when the experiments were tried ranged from 13° C. to 13.8° C., and for the present purpose may be regarded as constant. The lower or reading scale had forty divisions to the inch, and was distant from the glass fibre and mirror 38 inches, excepting in Experiment V, when it was at 37 inches. Sufficient time elapsed between the experiments to allow all sign of change due to after-effect of torsion to disappear. In all cases the first line of the table gives the time in minutes from release from torsion, the second the deflection of the image from its initial position in scale divisions. t Experiment I.-The twisting lasted 1 minute. Scale divisions .. t. t 1 2 3 4 5 7 10 17 25 22 13 9 7 5 4 3 2 1 Experiment II.-The twisting lasted 2 minutes. 1 2 3 4 5 7 10 20 40 Scale divisions.. 38 25 18 15 13 10 8 4 Experiment III.-Twisted for 5 minutes. 1 2 3 4 5 3 31 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 {y (T+t)-y (t)} where (t) = fp (t) dt. If now T = t1 + t2 + t3 + .. we may express the effect of one long twist in terms of several shorter twists by simply noticing that X{y(t)—y(t+T)}=X[{y(t)−− y(t+ts)} + {y(t + t1) − y(t + t1 +ts)} + {y(t + t1 + t2) − y(t + t1 + t2 + ts)}+, &c.] Apply this to the preceding results, calculating each experment from its predecessor. Let be the value of y (T+t)−y (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. Results for T=5 compared with those from T=2 and T=1. t a observed.... calculated... 1 2 observed.... 087 072 036 010 2 calculated... 066 047 Results for T=10 compared with those from T=5. Results for T-20 compared with those from T=10. Results for T=121 compared with those from T=20. 18 Xt t 7 648 612 556 830 780 730 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 subject 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 ø(†)=4, we have at time t after termination of a twist lasting time T, xt=A{ log(T+t)—log t}, the logarithms being taken to any base we please. The results were T+t t plotted on paper, «, 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 deviation appears to indicate the form (t), a being less than, but A ta near to, unity. If a=0.95 we have a fairly satisfactory formula. xt=AT where A'= when T-121. A ta In the following Table the observed and calculated values of ≈ when T=121 are compared, A' being taken as 0·032. ≈ observed.... 497 433 325 212 174 144 18 2 calculated... 493 429 320 218 176 147 42 To show the fact that A' decreases as T increases if a be assumed constant, I add a comparison when T=20, it being then necessary to take A'=0·037. t 1 2 3 4 5 7 10 x observed...、 0·00580 470 398 358 327 276 234 * calculated,,, 0.00607 485 422 370 337 285 233 15 25 40 60 80 100 æ observed,... 188 140 111 085 072 066 185 125 089 067 052 041 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 (t)= these experiments give values of A ranging from 0·0017 to 0.0022. Boltzmann for a fibre, probably of a quite different composition, gives numbers from 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)=4. 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. A ta IV. "Note in correction of an Error in the Rev. Dr. Haughton's Paper Notes on Physical Geology. No. V" ("Proc. Roy. Soc.," vol. xxvii, p. 447). By the Rev. SAMUEL HAUGHTON, M.D., Professor of Geology in the University of Dublin, F.R.S. Received October 9, 1878. 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 geometrically. Let yr, 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 ya and y'a', are normals to the cycloidal arc yy'; it is well known that yC, (radius of curvature) is double y (chord of generating circle) or double y'; therefore the angle yay' is double the angle yCy'; but yay' measures the angular velocity of the wabble, when a is supposed at rest; therefore the angular velocity of ye is only half that of the wabble, if the axis of figure were at rest. Hence in 305 days, ya will turn through 180° only, and not 360°. |