have been assigned. So the order of the group is the product of three numbers, viz., the number of vertices, the number of edges containing a given vertex, and the number of faces passing through a given edge. 4. For the six regular cells of S4 the results are:

5. Remarks.—(a) A deeper study proves the group of the five-cell to be holohedrically isomorph with that of the icosahedron.

(b) In the pairs of cases (C, C18) and (C120, Coo) the results are equal. This is due to the fact that these pairs of regular cells are reciprocal polars of each other with respect to a hypersphere.

(c) The order of the group is equal to 2r times the number of faces, r representing the number of vertices situated in any face.

Five-dimensional Space (S3).

6. General Principle extended.-The order of the group is the product of four numbers, viz., the number of vertices, the number of edges through a given vertex, the number of faces through a given edge, and the number of limiting bodies adjacent at a given face.

7. Results:

8. Remarks. (a) The cases (B) and (B) are reciprocal polars of each other, &c.

(b) The order of the group is equal to 6r times the number of limiting bodies, representing the number of vertices situated in any limiting body.

10. Remarks.—(a) The cases (B) and (B) are reciprocal polars of each other, &c.

(b) The order of the group is equal to (n-2)!r times the number of limiting beings of n-2 dimensions, representing the number of vertices situated in each of these.

8. On Mersenne's Numbers. By Lieut.-Colonel ALLAN CUNNINGHAM, R.E., Fellow of King's College, London.

These are numbers of form N=2-1, where p is prime. Lucas has shown that N is composite, and contains the factor (2p+ 1) when p and (2p+1) are both prime, and p is of form (41 + 3).

Such numbers N may for shortness be called Lucasians. The highest Lucasians, determinable by the existing tables of primes (extending to 9,000,000), are given by

p = 4,499,591 and 4,499,783,

and these are the only values of p yielding Lucasians in the range of 500 numbers between 4,499,500 and 4,500,000. An interesting group is given by

p=23+3=11; p=27+3=131; p= 215 + 3 = 32,771;

and these are the only numbers of form (2+3) yielding Lucasians when a not>26. Higher values go beyond the tables of primes.

Complete list of primes p of form (4+3), with (2p+1) also prime, when p not >2,500; these all give composites for N, and (2p+1) is a factor of N.

p = 11, 23, 83, 131, 179, 191, 239, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 911, 1,019, 1,031, 1,103, 1,223, 1,439, 1,451, 1,499, 1,511, 1,559, 1,583, 1,811, 1,931, 2,003, 2,039, 2,063, 2,339, 2,351, 2,399, 2,459.

It seems probable that primes of one of forms p= (2+1), (2+3) will, with exception of those yielding Lucasians, generally yield prime values of N, and that no others will; all the known (and conjectured) prime Mersenne's numbers fall under this rule.

9. End Games at Chess. By Lieut.-Colonel ALLAN CUNNINGHAM, R.E., Fellow of King's College, London.

Investigation of the number of positions in all the end games' at chess when there are only two or three pieces on the board. The results are:

10. Experiments showing the Boiling of Water in an open Tube.
By Professor OSBORNE REYNOLDS, F.R.S.

11. Report of the Committee on Earth Tremors.-See Reports, p. 145.

12. Report of the Committee on Meteorological Photography.-
See Reports, p. 143.

13. Report of the Committee on Solar Radiation.-See Reports, p. 106.

14. Report of the Committee on Underground Temperature.
See Reports, p. 107.

15. Report of the Ben Nevis Committee.-See Reports, p. 108.

16. On Recent Researches in the Infra-Red Spectrum.
By Dr. S. P. LANGLEY.

This paper was ordered by the General Committee to be printed in extenso.-See Reports, p. 465.

17. A new Determination of the Ratio of the Specific Heats of certain Gases. By O. LUMMER and E. PRINGSHEIM.1

When a perfect gas expands adiabatically from the pressure p, to the pressure Pa, while its absolute temperature decreases from T, to T, we have

where T means the ratio of the two specific heats. Therefore T can be determined by four corresponding values of P1, P2, T1, and T. For this purpose we experimented in the following way.

A copper balloon, nearly globular, containing about 90 litres was placed in a bath of water whose temperature was maintained constant within 001 C. and could be measured by a thermometer. Let this temperature be T1. The balloon was filled with a gas, well dried and pure, which was compressed to the pressure P1, measured by a manometer of sulphuric acid communicating with the balloon. Then the gas was allowed to escape into the atmosphere through an aperture of the balloon. So it expanded to the atmospheric pressure P, given by the barometer. In this way the quantities P1, P2, and T, can be found easily. The only difficulty is to determine the temperature T, of the gas at the moment when the expansion is finished and the pressure has attained the value p. For this purpose we need a thermometer showing the variable temperature of the gas instantaneously, that is, a thermometer of a negligible small mass.

A thermometer of the required qualities was formed by a strip of platinum of an extremely small thickness, which is soldered at both ends to two copper wires insulated from each other and introduced air-tight into the balloon. From these the strip hangs down freely in the middle of the balloon. The strip of platinum with its conducting wires formed one arm of a Wheatstone bridge, so that we were able to measure its electrical resistance, and hence its temperature at every moment. The strip was prepared by the method first used for wires by Wollaston, and adopted for strips by Lummer and Kurlbaum. It was cut out of a platinum silver plate composed of a platinum plate with the thickness 06μ and a silver plate Gu thick. The middle part of the strip had a length of 10 cm. and a breadth of about 0.2 mm., while the two ends were formed by conducting laps 1.5 cm. long and about 4 mm. broad. The silver was removed by nitric acid only in the middle narrow part, so that the resistance of the conducting laps perfectly disappeared in comparison with that of the middle part. But also these conducting laps are thin enough to take at a short distance from the ends the temperature of the surround

The original source of publication is the Smithsonian Institution in Washington, which kindly granted liberal assistance from the Hodgkins Fund for carrying on this investigation.

ing gas, as is easily proved by calculation. In this way the fall of temperature at the ends of the strip is perfectly eliminated. The resistance of the strip amounted to 87 ohms at the temperature of 17° C.; the variations measured by us were between 1 and 4 ohms.

In order to find the lowest temperature T, reached by the gas while expanding we have only to measure the smallest value which the resistance of the strip takes during the expansion. We worked in the following way. First, when the balloon is filled with compressed gas of the temperature T,, the Wheatstone bridge is equilibrated so that no current is going through the galvanometer. This state we call the first equilibrium. Then the galvanometer arm is opened and out of the arm of the bridge opposite to the strip a part of the resistance is taken away, so that the resistance of this arm now is lower than that of the strip. Now the gas is allowed to escape out of the balloon, the temperature of the gas is lowered, and the resistance of the strip decreases and approaches that of the opposite arm. Immediately after the end of the expansion, at the moment when the strip has the smallest resistance, we close the galvanometer circuit, and now the galvanometer shows by its deflection if the resistance of the strip at this moment is higher or lower than the resistance of the other arm. Only in the case when both resistances are exactly equal does the galvanometer remain at rest. This state we call the second equilibrium. This second equilibrium is always to be attained by a systematic variation of the initial pressure of the gas, and when it occurs the galvanometer remains at rest for some seconds, showing that during this time there is no appreciable conduction of heat to the gas surrounding the platinum strip.

In this way four corresponding values of P1, P2, T1, and T, are found.

A small error in these experiments is caused by the heat radiating from the walls of the balloon to the strip cooled by the expanding gas, but it is possible to determine the amount of this error by experiment. For this purpose we executed the experiments described above, once with a simple platinum strip, the second time with the same strip after having blackened it by platinum black. In the second case the error due to radiation is increased in the same proportion, as the absorbing power of blackened platinum is larger than that of the uncovered, and therefore we found a smaller value of T than before. The relation of the two absorbing powers was determined by special experiments, and so we found the correction necessary to remove the error resulting from radiation.

The gases on which we worked were atmospheric air, oxygen, carbonic acid, and hydrogen.

The results obtained with the naked strip were:

To these values we must add the correction caused by radiation:

That we found a much greater value for hydrogen than all former experimenters is the best proof of the superiority of our method. All other methods failed with hydrogen because the heat conduction of this gas is very great, and consequently the experiments were not adiabatic. Our method gives also for hydrogen the second equilibrium of the Wheatstone bridge lasting for more than a whole second, showing that also here the expansion is quite adiabatic.

DEPARTMENT III.

18. A Method for accurately Determining the Freezing-point of Aqueous Solutions which freeze at Temperatures just below 0° C. By the Late P. B. LEWIS. Communicated by Dr. MEJER WILDERMANN.

Cover.

The thermometers used for measuring minute variations in temperature were graduated, the one to read to thousandths the other to hundredths of a degree. The total volume of the solution used in the various determinations was 1,250 c.c. The beaker containing the liquid for examination was placed on a felt cushion, wrapped in thin gutta-percha, inside a zinc vessel which acted as an air-chamber. The cover of this vessel was provided with a circular opening into which fitted a cork, holding the thermometers. The stirring of the liquid was effected by a porcelain stirrer which consisted of two parallel porcelain plates, in which there were corresponding holes for the thermometers to pass through, and besides these eight round holes so arranged that the holes in the upper plate did not correspond with those in the lower plate. On moving the stirrer up and down currents are sent in all directions. The thermometers are kept uniformly at a temperature of 0° C., even when not in use, by keeping them plunged in a glass beaker, filled with ice and water at 0° C. This second beaker is contained in a similar zinc vessel, which acts as a cold-air chamber. The two zinc vessels, the one containing the distilled water and the other the solution to be examined, are placed in a larger zinc vessel, which acts as an ice-bath, and which is encased in a yet larger zinc vessel, so that there should be a cushion of air of some centimetres' thickness between the two. The outermost zinc vessel stands on a thick felt mat, and is wrapped in a thick felt The ice-bath was covered over as far as possible with pieces of asbestos cloth. By these means a fairly constant temperature is maintained in the bath. The temperature of the ice-bath should only vary between 1°.8 and -2° when the temperature of the room is 12° to 18°, and only between-2°1 and 2°.2 at a temperature of 25°. It is necessary for a successful experiment to overcool the solution by about 0°7 to 1°. Under these conditions the most accurate determination of the freezing-point can be obtained: the mercury in the large thermometer has sufficient time to become exactly of the temperature of the solution, the ice is obtained in the form of thin films, and the freezing-point remains fairly constant for from ten to fifteen minutes, though slight variations of from 0° 0001 to 0°-0002 occur, and in the more concentrated solutions of 0° 0003. As regards the action of the thermometers it is important to know how far the capacity of their bulbs is affected by the pressure caused by a rise, especially a rapid rise, of mercury in the stem, and how quickly the bulb recovers from the expansion or contraction which variations of pressure or temperature cause. It can only be here stated that if such variations exist they lie within the limits of ordinary experimental error, namely, 0°-0001, 0°0002, and rarely 0° 0003. On the other hand, it was found that when the more delicate thermometer had been allowed to remain for several hours at the ordinary temperature, a rise in its indication of the freezing-point could be observed for many days afterwards, and a long-continued exposure to a rise or fall of atmospheric pressure causes a variation of about 0°-0003 in the reading of the freezing-point for 1 mm. rise of the barometer. A similar variation is produced in the reading of the smaller thermometer. In the case of very dilute solutions only those determinations of the freezing-point which have been made during almost constant conditions of atmospheric pressure are to be depended on. While the column of mercury is rising the stirring must be maintained together with a light tapping with the fingers on the cork. The maximum is reached when further tapping causes no rise of the mercurial column. When this point has been reached readings are made every minute for from eight to ten minutes. The readings are made with a small lens. After the constant point has been observed for from eight to ten minutes the form taken by the ice produced is examined. If the experiment has been successful the ice will have formed in fine films and in considerable quantity throughout the solution. An account of this investigation will be given in the Journal of the Chemical Society, and in the 'Zeitsch. f. physik. Chemie.'