Regnault gives 1200 77 and 1166.67, but these numbers are probably too small.

According to a few experiments which I made later with coal-gas, the form of the apparatus that was used for carbonic acid seems also suitable for gases lighter than air. In this case, however, the observer must operate as rapidly as possible.

§ 6. Mixtures.

I also applied the method with good success to mixtures of air and vapours. The form of the apparatus was the same as for air, only that in this case the bottle was filled with the evaporating liquid instead of with water.

I raised the liquid in the tube as far as the side-piece A, then let it slowly sink and remain standing for a time, which varied from a few seconds to two hours. The temperature for all the experiments was constant (17° C.). With tube II. and fork I took six sets of readings given below. At the beginning of each set, one or two half wave-lengths come where only very little vapour was present. Then with increasing saturation the half wave-lengths gradually decrease, until in the immediate neighbourhood of the liquid they

become constant.

The mean of the last six half wave-lengths in the immediate neighbourhood of the surface of the liquid is 109.5 millim. The velocity of sound in air v is calculated by the formula

where B denotes the barometric height, Q the specific gravity of mercury, g the accelerating force of gravity, o the density of air referred to water, and k the ratio of the specific heats. The analogous formula, when vapour is present, is

where udensity of the vapour compared with air, and a=thermal coefficient of expansion of gases.

Or

we get as ratio of the specific heats of ethyl-ether

k=1.0202.

Jaeger* found 1.097 (at 20°) and Müller† 1.0288 (between 45°-4 and 22°-5).

A phenomenon, similar to the one observed in a mixture of air and vapour, I found also in a mixture of air and carbonic acid.

In the manner already described I filled the tube half full with carbonic acid, then turned the gas off and let the water sink. The upper portion of the tube was thus filled with air and the lower with carbonic acid. With fork cand tube II. I found the following readings:

In both columns the two first half wave-lengths agree with those previously found for pure air, the two last with those for pure carbonic acid. This is a simple method of ascertaining with fair exactness the relative velocities of sound in air and carbonic acid or other suitable gases.

* Wied. Ann. xxxvi. p. 209 (1889).
† Wied. Ann. xviii. p. 116 (1883).

§ 7.

After my experiments were finished I had some doubt as to whether the vibration-frequency of c, was exactly 256. I decided the point by means of an electric registration method. A small, thin, pointed piece of platinum-plate was attached to the end of one of the prongs of the fork, and the vibrationcurves were traced on a metal cylinder coated with blackened paper. A weak induction-current, with mercury contact with the seconds-pendulum of a clock of known daily error, was generated and conducted by the platinum point to the blackened paper. The number of waves between the marks of the sparks, taken two and two, gave the double vibrationfrequency of the fork. As a result I found that e, made 256 23 vibrations per second. This necessitated small corrections in my results, which are shown below.

* Mém. de l'Inst. xxxvii. p. 133; Compt. Rend. lxvi. p. 219 (1868).

Jaeger, Wied. Ann. xxxvi. p. 209 (1889).

Pogg. Ann. cxlviii. p. 606 (1873).

& Wied. Ann. xviii. p. 116 (1880).

The experiments with hydrogen can only be regarded as approximately correct.

My results for the velocity of sound in air and carbonic acid in glass tubes of different diameters are in full agreement with Kirchhoff's theoretical formula founded upon the consideration of the friction and the conduction of heat of gases.

The velocity of sound in free space for air and carbonic acid is, according to my results, invariable for tones of different pitch and intensity.

It is my pleasant duty to express here to Professor Quincke my heartiest thanks for his kind support and instructive counsel during the prosecution of the above inquiry.

The Physical Institute, Heidelberg,
January 1894.

XXIX. The Hatchet Planimeter. By F. W. Hill*. THE hatchet planimeter consists essentially of a tracingpoint and a convex chisel-edge rigidly connected, the point and the edge being in the same plane. When the point is moved along any line, the edge describes a curve of pursuit.

The object of this paper is to investigate how the instrument may be used to determine areas.

Let the tracer start from a point O inside the area, move along any line to the perimeter, then round the perimeter and back along the same line to O; the solution of the problem consists in finding an expression for the angle AOB between the initial and final positions OA, OB of the line joining the tracing-point and chisel-edge.

All attempts to express the area of the curve in terms of this angle proved futile except in a few special cases, such as the circle and square; but the magnitude of the angle may be found in the form of an infinite series, the most important term of which is a multiple of the area. The complexity of the result would seem to show that no simple geometrical explanation is possible.

Let the tracer move a distance r along a straight line (fig. 1); then, if x, x' be the initial and final inclinations of the rod to the line, e the length of the rod, it is easily proved that

* Communicated by the Physical Society: read June 22, 1894.

Phil. Mag. S. 5. Vol. 38. No. 232. Sept. 1894.

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