infinite it is finite, as shown by (16). Further, we observe that the amplitude in the case of an elastic hammer is greater than that due to a hard hammer.

III. Experiments.

The hammers used in the piano in the upper octaves are harder and lighter than those used in the middle ranges. The distance of the striking-point is almost the same for all strings; it varies within one-ninth to one-seventh of the length of the string in different types of piano. Hence, in order to test the above formulæ, a sonometer was fixed almost vertically and the hammer was placed near the lower extremity of the sonometer, so that the distance of the point struck from the lower bridge was small. By an arrangement of a lever the hammer was made to strike in a vertical plane-that is, parallel to the plane containing the hammer and the string. The lever arrangement was the same as is used in the piano. The wires and hammers used in the experiments were borrowed from a local pianotuner, and those were selected which are used in the middle range where the hammer is elastic and heavy. The point of impact was photographed by a sliding plate which recorded the approach and retreat of the hammer. (Care was taken that the hammer did not strike the string a second time, by an arrangement of a catch.) The time was simultaneously obtained from the trace of a style attached to a tuning-fork. In Pl. VIII. figs. 1, 2, 3, 4, and 5 show in ascending order the distance of the point struck. The approach and retreat of the hammer are shown by the parabolic shadow, and the vibration of the point struck and the record of the style are also seen. It is evident from these figures that the duration of impact and the amplitude of the resulting motion increase with increasing distance of the striking-point. These facts are in agreement with those deduced from (14) and (15). Formula (14) can be written in the form:

0 =

Duration of impact.

Period of vibration of the string.

Table I. gives a record of the values at different distances a found experimentally and calculated from (14).

M = 7.3 grams (mass of head plus one-third mass of stem).

From the table it is clear that the values observed and those calculated from (14) agree fairly, while Kaufmann's formula gives more divergent values than experimental errors justify. Further, it is also confirmed that the duration of impact does not depend upon the velocity of impact, to which no fixed value was imposed during experimentation.

Hence it is concluded that to determine the law of pressure between the hammer and the string, its elasticity must be taken into consideration. In the higher scale of the piano, where the hammer is hard and light, Kaufmann's theory of inelastic impact may hold good. It is also found that the calculated values of 4/0 differ from those found experimentally with increasing striking distance. They are less than those observed. This regular decrease in the calculated values points out that our assumption, namely that the portion of the string between the striking-point and the nearer extremity follows the motion of the hammer, does not hold true when a is large, appreciable time being taken by the wave to reach the nearer end and return.

Conclusion.

Since in practice the striking distance is about one-ninth the length of the string, it is thought that the above theory

of the elastic hammer will be of practical value. It is more general than Kaufmann's, and includes his as a special case. The authors hope to calculate the intensity of partials on the basis of the law of pressure given by them, and compare it with experiments and verify formula (15) with different values of o on a future occasion.

Physics Laboratory, Allahabad University,

Allahabad, India,

December 14th, 1922.

CX. The Relation between Uranium and Radium. Part VIII. The Period of Ionium and the IoniumThorium Ratio in Colorado Carnotite and Joachimsthal Pitchblende. By FREDERICK SODDY, M.A., F.R.S., and Miss ADA F. R. HITCHINS, B.Sc.*

N the course of a study of the ionium-thorium ratio in minerals, occasion has been taken to redetermine the period of ionium from the rate of growth of radium in the various uranium preparations purified from 18 to 15 years ago, and the new results may first be recorded. It may be recalled that, as the result of the last examination in 1919 (Phil. Mag. [6] xxxviii. p. 483 (1919)), it was found that the rate of growth of radium was proceeding regularly in accordance with the square of the time, with the period of 237,500,000 years as the product of the periods of average life of ionium and radium, which led to the adoption of the period of 100,000 years for that of ionium. The new results are set forth in the same tabular form as in the paper cited. The value of the average life of radium is retained at 2375 years for the sake of uniformity, and the average life of ionium, 1/2, is calculated from the formula

1/λ2 = (U/Ra)(3·4 × 10-7/2375)T2,

where Ra grams of radium are produced from U grams of uranium in T years.

Two sets of determinations have been made, in August and December 1923, of the amount of radium in the four uranium preparations. The electroscope was first calibrated with a number of radium standards, made from a radium barium chloride of known small radium content, one of which was new and the others those previously employed.

* Communicated by the Authors.