the summation including all the successive values of d at the successive times dt Similarly, for the second linear function we take 12. Then we may make dt infinitely small and N infinitely great, while Nôt = T and our linear functions become 19 or §1-X1, §2-X2, &c., if X,, X,, &c., now denote the initial values of 1, §1, &c. And our proposition now assumes the form that the chance of 1 . . . §n lying within assigned limits varies as where the index is a quadratic function of (§, - X1), (§2 − X2), &c. We thus get the exponential form, or 'law of error,' in all cases for which the modified condition of independence is satisfied. In order that it may be satisfied the time variation of §, . . . §n must depend not only on the values of §,... En for the time being, but also, in some appreciable degree, on external fortuitous causes. And the greater the comparative importance of these fortuitous causes, the shorter is the time T which, as the interval between two successive states of the system, makes the variables in one of those states independent of those in the other. Department II. METEOROLOGY. 1. Probable Projection Lightning Flashes. The classification of lightning flashes is already beset with difficulties. The object of this paper is to suggest the possibility of Projection Lightning Flashes, the existence of which would increase the difficulties of classification. Sheet lightning reflected on clouds is made up of numerous images of the lightning flash superimposed one upon the other. If there is a cloud with a sufficiently small opening in it between the lightning discharge and a reflecting surface of clouds, the latter being in an adequate position, on the clouds that otherwise would have been merely illuminated with sheet lightning will appear the optical projection of a lightning flash. If the surface of clouds upon which the image falls is level, the image will be a perfect reproduction of the lightning flash, save that it is inverted, and to some extent dulled in brilliancy. If, however, the surface of the clouds is irregular, such as those of the cumulus type, the image of the flash will take the shape of the irregular surface. In this way a zig-zagged flash with long angles could be formed very like the lightning flashes depicted by artists. If there happen to be more openings than one in the clouds between the flash and receiving surface, there will be a corresponding reduplication of the flashes, which may perhaps explain some of the multiple effects observed. If projection flashes occur in nature it becomes a question whether the photographic plate can register them. 2. Report on Solar Radiation.-See Reports, p. 81. 3. Report on Earth Tremors.-See Reports, p. 184 4. Reports on Earthquakes in Japan.-See Reports, pp. 81, 113. 5. On some Experiments made with Lord Kelvin's Portable Electrometer. By ARTHUR SCHUSTER, F.R.S. Experiments made during a recent trip to Switzerland have demonstrated the great convenience of Lord Kelvin's portable instrument for the observation of atmospheric electricity. The electrometer can easily be carried in a knapsack to almost any place the mountaineer can reach, and the observations are readily performed in a short space of time. The object of the experiments was to learn, if possible, whether the electric force at great heights is sensibly different from that at the sea-level. But owing to the difficulty of comparing together observations made in valleys with those made on mountain peaks, the measurements made during the past year were entirely of a tentative character, more for the purpose of testing the working of the instrument than of obtaining any definite results. The numbers given are only relative, and are referred to an arbitrary unit. Numerous experiments made in fine weather in the centre of a small meadow in front of the hotel at Pontresina gave for the electric force, on the average, about 30, while the number obtained on September 13, in Ipswich, in a place rather more sheltered by surrounding houses, was between 80 and 100.1 On the so-called 'Diavolezza Pass' the force was as high as 147; on the Isola Pers, a rocky island, projecting well out of the surrounding glaciers, the number found was 116. The instrument was taken on August 19 to the top of the Glüscheint (12,000 feet), which consists of a ridge of rocks just wide enough for one man to stand upon. The electric force reached the number 816, while on the snowfield a few hundred feet below the top, the force was only 170. The top of the Schafberg gave 357. On the Morteratsch glacier the numbers were, as a rule, very low, except on one occasion, when a strong downward wind was blowing; even then the force was only 42. Half a mile below the foot of the glacier, the numbers were very irregular, and seemed to depend a good deal on the direction of the wind. On one occasion, when that direction was frequently changing, a negative electrification was observed whenever the current of air was away from the glacier. This may have been due to the negative electrification caused by the glacier stream in a way similar to that in which a similar electrification is produced by a waterfall. The diurnal variation was observed on one day by taking measurements every half-hour during the morning and every hour during the afternoon till seven o'clock. A maximum of 36 was observed at 11 A.M. The numbers then fell In Manchester, during the fine weather in the last fortnight of September, the numbers went up to 250. 1895. SS irregularly till four o'clock, when the record was only 14. After that a rise took place (31 at 5 P.M. and 29 at 6 P.M.). At 7 P.M. no sensible force could be measured at all. The wind, which had been blowing up the valley all day, changed in direction between six and seven o'clock, and was blowing down the valley when the observations had to be discontinued, owing to want of light. The author does not draw any conclusions from these observations, but does not believe that a mere difference in the configuration of the ground is sufficient to account for the great differences in the electric force observed. 6. On Indian Thunderstorms. By C. MICHIE SMITH. Observation shows that the ordinary hot-weather displays of sheet lightning take place in the region between the sea aud land breezes where there are wellmarked ascending currents, as shown by the great masses of pillared cumulus cloud. These clouds are usually in pairs, and much of the sheet lightning consists of discharges taking place between the two clouds forming such a pair. The land and sea breezes differ from each other in dryness and in dustiness, and many observations seem to show that thunderstorms are originated only where dry, dusty, and moist, and comparatively dustless air-currents meet. This explains why electrical phenomena are observed only in that part of a cyclone in which the air from the sea meets the air from the land, and why 'nor'-westers' are accompanied by such brilliant electrical displays. Dry, dusty air is usually negatively electrified relatively to the earth, while the sea breeze is usually positively electrified. The electrical displays take place in clouds that are rapidly sinking, and such clouds are often surrounded by an iridescent fringe (the colours of which are due probably to dust and moisture, as explained by Aitken, composed of the smaller particles left behind by the sinking cloud). 7. On the Zodiacal Light considered as an Atmospheric Phenomenon. By W. H. WOOD. 8. On the Local Origin of the Aurora Borealis. By W. H. WOOD. 9. Report on the Application of Photography to Meteorology. 10. Report on the Meteorological Observations on Ben Nevis. MONDAY, SEPTEMBER 16. 1. A discussion on the Objective Character of Combination Tones' was introduced by the following Paper : Notes on the Objective Existence of Combination Tones.1 By A. W. RÜCKER, M.A., F.R.S. It might at first sight appear that the question of the objective existence of combination tones would depend upon two others—viz. Are such notes heard? Do they exist as pulses in the air? In the following notes the references to statements of Von Helmholtz are for convenience made to the second English edition of the Sensations of Tone, translated by Ellis. The references to König's views are to Quelques Expériences d'Acoustique. Paris, 1882. The matter, however, is not so simple. All agree that notes corresponding to the difference tone are heard under some circumstances, but many deny that they are produced as Von Helmholtz supposed, and would therefore deny that they are combination tones. Again, Von Helmholtz, who was the most prominent supporter of the objective reality of these notes, was also the author of the theory which explains their production within the ear itself. It is therefore better to begin with the second question. Do notes corresponding in frequency with the combination tones accompany the two fundamental notes as air-waves under any circumstances? The physical evidence for and against an affirmative answer is as following:Von Helmholtz stated that he had set membranes in motion by combination tones produced by the siren, and air resonators in motion by combination tones produced by the harmonium (Ellis, p. 157). I am not aware that the experiment with membranes has been repeated in the same form, but O. Lummer (' Verh. Phys. Gesell., Berlin, 1886, No. 9, p. 66) claimed to have detected the tones by means of the microphone. On the other hand, König (p. 130) denies that combination tones are reinforced by resonators; and Bosanquet satisfied himself that the ordinary first difference tone is incapable of exciting a resonator. More recently, the writer and Mr. Edser, using as a resonator a tuning-fork of frequency 64, the motions of which were detected by attaching to one of the prongs a mirror which formed one of a system by which Michelson's interference bands were produced, have obtained evidence of the objective character of the summation and difference tones produced by a siren. They have confirmed these results in the case of the summation tone by a Rayleigh vane-resonator as modified by Boys (Phil. Mag.,' April 1895, p. 341). The only objection which, as far as the writer knows, has been brought against these experiments is that the tones detected must be of very small intensity; and Mr. Bosanquet has stated (in a letter) that he does not wish to be understood as denying the existence of very feeble combination tones. It is unnecessary to quote experiments made by various observers with tuningforks, as the use of these instruments is in general opposed to the directions and theories of Helmholtz. (Ellis, pp. 157, 158.) On the whole, then, the evidence appears to be in favour of the view that objective notes of the same frequency as the combination tones do exist, at all events in special cases. Their relative intensity to the fundamental notes has not been determined, and is probably small. Turning next to questions of theory, three explanations have been given of objective combination tones-viz., that they are due (1) to beats, (2) to finite displacements of the vibrating particles, and (3) to intermittence. Among other objections to the first theory, Helmholtz pointed out that it would not explain the summation tone. (Ellis, p. 156). Hence König suggested that the summation tone might be due to the beats of partials (König, p. 126). This explanation requires that, if p and q are the frequencies of the fundamentals, p+q=n (p−q) where n is an integer. The writer and Mr. Edser have, however, obtained evidence of the objective existence of the summation tone when p/q=16/9, so that n=25/7 (loc. cit. p. 352). (See Ellis, p. 530.) Appunn and Preyer have suggested that the summation tone is the beat tone between the first partial of the higher note and the difference tone, for 2p−(p −q)=p+q. König (p. 127) strongly opposes the adequacy of this explana tion, which is contrary to his own observations on beats, and which fails to explain why the difference tone should not produce equally permanent effects by beating with the first partial of the lower note, thus giving 2q~(p-2)=3q-p or p-3q. The theory of finite displacements is due to Helmholtz, who has shown (Ellis, p. 412) that if the elastic forces are not symmetrical about the position of equilibrium, the fundamental tones will be accompanied by the second partials and the difference and summation tones. He has, however, also proved (Ellis, p. 420) that if in an instrument such as the siren the opening of one hole affects the pressure under which the air is simul taneously escaping out of the other, the quantity of air escaping may be represented in the simplest case by such a formula as Q=C(1-sin 2 m2) (1 − sin 2π nt) = C [1-sin 2 mt - sin 2 nt + {cos 2ñ (m—n) t — cos 2′′ (m +n) t} ], thus giving rise to the first difference and summation tones. A somewhat similar theory has since been elaborated by Terquem. ('Annales d'Ecole Normale,' 1870, p. 356). At present it must be considered to be open to question whether the objective combination tones given by the siren are due to finite displacement or to intermittence, or to both causes combined. 2. A Discussion on a New Practical Heat Standard.' Introduced by a Paper by E. H. Griffiths, F.R.S. 3. On the Thermal Conductivities of Mixtures of Liquids. The author has carried out a number of determinations of the thermal conductivities of mixtures of liquids by a method analogous to that of Christiansen, but with the heat supplied electrically and the temperature measured by means of thermo-junctions. For the mixtures of water, glycerine and alcohols experimented on, the conductivities are found to be less than the values calculated from the amounts and thermal conductivities of the constituents, and the author believes that this will be found to be a general law. A possible explanation may be found in the inability of one of the molecules of a mixture to take up and transmit the particular kind of vibration executed by the other. For solutions of salts and gases in water the author finds conductivities less than that of water, in agreement with the results obtained by Jäger. 4. A Method of Comparing the Heats of Evaporation of Different Liquids at their Boiling Points. By Professor W. RAMSAY, Ph.D., F.R.S., University College, London, and Miss DOROTHY MARSHALL, B.Sc., University College, London. This method consists in making the liquid boil by passing an electric current through a wire immersed in it. The liquid is put into a glass bulb enclosed in an outer jacket filled with vapour of the same liquid. An open tube is attached to the top of the bulb, so that there is free communication between the interior and the vapour jacket, and no loss of material. Inside the bulb is a spiral of fine platinum wire, attached to stout platinum terminals which are sealed into the glass. These terminals rest in mercury cups, by means of which connection is made. The temperature of the liquid in the bulb is raised to the boiling-point by the vapour jacket; thus when a current is sent through the wire the whole of the heat developed is spent in converting a portion of the liquid into vapour. If two such bulbs containing different liquids are connected in series, the ratio of their losses of weight is the inverse ratio of the heats of evaporation of the liquids. A correction must be made for the inequality in resistance of the spirals. The ratio of the differences of potential between the ends of each spiral while the current is passing is determined in each experiment by Poggendorff's method, |