be anticipated, as wire No. 11 was not heated in the determination of the curve 5 in fig. 6. Finally, in this connexion it may be remarked that, as seen from fig. 6, the initial sensitivity of the arrangement employing three wires in each arm, as already explained, is practically identical with that of the two-wire arrangement. This initial decrease in the anticipated sensitivity is, of course, attributable to the fact that No. 6 wire, with its small initial rate of increase of temperature with increase of the impressed velocity, is included in the second group of wires constituting such a device.

The calibration curves given in fig. 7 were obtained by inserting the single wires indicated in the arms of the bridge, intermediate wires functioning as a heating coil. In the case of the full-line curves in fig. 7, all remaining wires of

the series were also heated, and in the case of the brokenline curves, wires subsequent to the pair inserted in the bridge were not heated. The heating current, both in the anemometer wires and in the heating coil, were throughout adjusted to 11 amp., and the galvanometer shunted with 10 ohms. The curves in fig. 7 are therefore, strictly comparable with those given in fig. 6. Considering first the full-line curves, which were obtained with all wires of the series heated, it will be seen that, while the ultimate sensitivity of the anemometer device for comparatively large velocities is greater, the later the wire employed in conjunction with the first is situated in the sequence of wires, yet this is not so for small values of the impressed velocity. Thus, whereas the respective deflexions corresponding to an impressed velocity of 8 cms. per sec. are 48, 150, and 200 when the 2nd, 5th, and 11th wires are used in the bridge in conjunction with the 1st wire, the deflexions corresponding to an impressed velocity of 4 cms. per sec. are respectively 32, 82, and 57. It is clearly seen that for low values of the impressed velocity the arrangement employing the 5th wire in conjunction with the 1st is more sensitive than that employing the 11th in conjunction with the 1st. The explanation of this phenomenon is clear from a consideration of the curves in fig. 3, wherein it will be seen that initially the temperature of No. 5 wire rises more rapidly than that of No. 11 wire, although, as seen from fig. 2, wire No. 11 attains the highest ultimate rise of temperature of the whole sequence of wire. It is clear from fig. 3 that, under the conditions specified, wire No. 5 used in conjunction with wire No. 1 affords the hot-wire anemometer employing one wire in each appropriate arm of the bridge possessing the maximum sensitivity in the region of very low velocities. From fig. 7 it will be likewise seen that, in accordance with anticipations advanced on page 249, the sensitivity of the anemometer device employing wires Nos. 1 and 2 is greater when subsequent wires are not heated than when such wires are heated. It will also be noticed that while this is also true of the anemometer device employing wire No. 5, in conjunction with wire No. 1, for values of the impressed velocity greater than about 6 cms. per sec., a reversal of the relative sensitivities occurs below this velocity, the greater sensitivity being then shown by the device in which subsequent wires are heated. The case of wires Nos. 1 and 5 being employed differs essentially from the case in which wires Nos. 1 and 2 are employed. As already pointed out, the extreme wires in a sequence are subject, as shown in figs. 4 and 5, to what may be termed an "end effect." The use

of wires Nos. 1 and 2 alone, results in what is tantamount to the abolition of the end effect, as the end effect in this case is the whole effect, there being only two wires. The effect of the subsequent heated wires upon the temperature rise experienced by No. 2 wire owing to an impressed air-stream moving with slow velocity has already been discussed. The case of wire No. 5 being employed in conjunction with wire No. 1 can be readily discussed as follows:- When wires subsequent to No. 5 in the sequence are not heated, wire No. 5 is one of the end pair of wires, and is subject to what has been termed the "end effect." Its initial temperature and that of No. 4 wire are both less than that of No. 3 wire. When wires subsequent to No. 5 in the sequence are heated, wires 4 and 5 no longer experience the end effect. Their temperatures are considerably higher than was previously the case. With the establishment of a slow impressed stream of air, wire No. 5 now experiences a thermal effect, due principally to the approach towards it of the hot convection current arising from No. 4 wire. The convection current being warmer than was previously the case, the rise of temperature of wire No. 5 is materially greater, and the sensitivity of the anemometer device employing wires Nos. 1 and 5 consequently initially greater when wires subsequent to No. 5 are heated than is the case when these are not heated electrically. Possibly the matter may be made clearer by reference to fig. 3. There it will be seen that initially the temperature rise experienced by wire No. 11 the last of the sequence, is very much less than that experienced by the adjacent wire No. 10. In like manner, it is to be anticipated that when No. 5 wire is the last of the sequence of heated wires, its rise of temperature due to an impressed air-stream will be initially small compared with what it would be if wire No. 6 were heated, and similarly for subsequent wires of the whole sequence. In fig. 2 it will also be seen that the last wire of the sequence ultimately attains the greatest rise of temperature. Similarly, it is to be anticipated that when wire No. 5 is the last heated wire of the sequence, with increase in the velocity of the impressed stream, its temperature rise will ultimately be greater than when subsequent wires of the sequence are heated. Under these circumstances it is to be expected that ultimately the sensitivity of the anemometer device employing wires Nos. 1 and 5 will be greater if subsequent wires in the series are not heated than would be the case if these latter were heated.

In conclusion, it may be remarked that the sequence of wires illustrated in fig. 1 may be used, after the manner Phil. Mag. S. 6. Vol. 41. No. 242. Feb. 1921.

S

employed in platinum thermometry, for purposes of anemometry. Thus wires Nos. 1 and 11 might be inserted in a bridge, employing a bridge current of, say, 001 amp. Intervening wires would be employed as a heating coil. The deflexion-velocity calibration curves obtained in this manner present the same main features as those discussed in the present paper. With increase in the impressed velocity, the deflexion increases until an upper limit is reached, and thereafter decreases in the manner already described. The sensitivity of the device is, however, owing to the small current employed in the arms of the bridge, very considerably smaller than that of the type of anemometer described in the main part of this paper. Thus, using the same electrical apparatus, with its sensitivity equal to that employed throughout this work, in conjunction with what may be termed the thermometric anemometer just described, the heating current being 11 amp., the maximum deflexion obtained was 10 scale divisions. The thermometric type of hot-wire instrument has been introduced by C. C. Thomas for the measurement of gas-flow. The electric energy requisite to maintain a constant difference of temperature of 2° C. in a pair of differential platinum thermometers situate one on each side of a heating-coil arranged across the section of the tube, is measured. It is clear that with certain dispositions of the thermometers and heating coil, for very small values of the velocity of the gas stream, the considerations advanced in the present paper become of importance. Thus, provided the heating coil and thermometers are suitably disposed, for low velocities of the gas stream, the electrical energy necessary to maintain a constant difference of temperature between the two thermometers may decrease with increase in the impressed velocity of the gas stream.

The work detailed herein was carried out in the Physical Laboratory of the South Metropolitan Gas Company. The author desires to express his sincere gratitude to Dr. Charles Carpenter, C.B.E., for his unfailing and inspiring interest in the research, and for the ready provision of all facilities necessary for the prosecution of the work.

Physical Laboratory,

South Metropolitan Gas Company,

709 Old Kent Koad, S.E.

20 Oct., 1920.

Journ. Franklin Inst. 1911, pp. 411-460; Trans. American Soc. Mech. Eng. 1909, p. 655; Proc. American Gas Inst. 1912, p. 339.

XXI. On the Period of Vibration of the Gravest Mode of a Thin Rod, in the form of a Truncated Wedge, when in Rotation about its Base*. By LORNA M. SWAIN, Lecturer in Mathematics, Newnham College, Cambridge.

THIS

HIS problem was suggested to the author, in connexion with work undertaken at the Royal Aircraft Establishment, under Mr. R. V. Southwell, to estimate the effect of centrifugal force on the periods of vibration of airscrew blades. The effect was worked out rigorously for the case of a uniform rod by Mr. Arthur Berry and for the case of a rod tapering to a knife-edge by Mr. H. A. Webb and the author§. As explained in the latter report, the mathematical work for a rod tapering to an arbitrary depth would appear to be somewhat involved, and it seemed worth while to consider if the problem could not be tackled by some other method, likely to give results sufficiently accurate to be of use.

This paper contains an attempt to apply Lord Rayleigh's method for the calculation of the period of vibration of the gravest mode to this case and also gives some estimation of the accuracy likely to be attained.

y

L

Fig. 1.

N

Consider a thin rod in the form of a truncated wedge, symmetrical about, an axis (oz), encastred at the base, which contains the origin o, and rotating with uniform angular velocity w about a perpendicular axis (oy). The breadth, measured perpendicular to the plane yoz, is uniform,

For the case of no rotation, see also a paper by J. Morrow in Phil. Mag. vol. x. p. 124 (1905).

† Communicated by the Author.

Advisory Committee for Aeronautics. Reports and Memoranda, No. 488.

$ Advisory Committee for Aeronautics. Reports and Memoranda, No. 626.

Rayleigh, Theory of Sound,' vol. i., §§ 88, 89.