Problem. If p be one of the roots of the equation xm − 1 = 0, (not 1,) then (1 − p)(1 − p2) …. (1 − pm − 1) = m, provided m is a prime number.

If m be not a prime number, and if , the same will hold for all roots p = p1α, where α is a number < m and prime to m. But for all roots p = p1α, where α, or one of its prime factors, is also a prime factor of m, the product (1 − p)(1 − p2) …. (1 − pm − 1) will be equal to 0.

1. The method for the solution of questions in the theory of probabilities applied in this paper, is that which was developed by the author in a treatise entitled, “An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities.” The practical object of the paper is to deduce from that method certain conclusions relating to the combination of testimonies or judgments. Beside this, however, it will have a speculative reference to some more general questions connected with the theory of probabilities; and especially to the following question, viz.: To what extent the different modes in which the human mind proceeds, in the estimation of probability, may be considered as mutually confirming each other,—as manifestations of a central unity of thought amid the diversity of the forms in which that unity is developed.

That there must be a medium forming a continuous material communication throughout space to the remotest visible body is a fundamental assumption in the undulatory Theory of Light. Whether or not this medium is (as appears to me most probable) a continuation of our own atmosphere, its existence is a fact that cannot be questioned, when the overwhelming evidence in favour of the undulatory theory is considered; and the investigation of its properties in every possible way becomes an object of the greatest interest. A first question would naturally occur, What is the absolute density of the luminiferous ether in any part of space? I am not aware of any attempt having hitherto been made to answer this question, and the present state of science does not in fact afford sufficient data. It has, however, occurred to me that we may assign an inferior limit to the density of the luminiferous medium in interplanetary space by considering the mechanical value of sunlight as deduced in preceding communications to the Royal Society from Pouillet's data on solar radiation, and Joule's mechanical equivalent of the thermal unit.

The following paper contains the details of some experiments upon the action of ammonia on the oils and fats, of which a preliminary notice was published in the Quarterly Journal of the Chemical Society of London. The production of a soapy emulsion by the action of ammonia on these substances has long been familiar to chemists, but comparatively few accurate experiments have been made upon the compounds formed. Boullay long since examined the crystalline substance obtained from olive-oil, which he called Margaramide, and mentioned that similar compounds were obtained with the other oils, although he did not examine them. More lately, M. Bouis produced ricinolamide from castor-oil, and showed that, by fusion with potash, it yielded caprylic alcohol and sebacic acid; and still more recently, he has obtained another fatty amide, which he calls Isocetamide, by the action of ammonia on the fat of the purging nut of the West Indies.

The mutual actions and motions of the heavenly bodies have long been regarded as the grandest phenomena of mechanical energy in nature. Their light has been seen, and their heat has been felt, without the slightest suspicion that we had thus a direct perception of mechanical energy at all. Even after it has been shewn that the almost inconceivably minute fraction of the Sun's heat and light reaching the earth is the source of energy from which all the mechanical actions of organic life, and nearly every motion of inorganic nature at its surface, are derived, the energy of this source has been scarcely thought of as a development of mechanical power.

The phenomena presented by the prismatic spectra of flames have occupied the attention of many and excellent investigators. In most instances, however, no attempt has been made to procure accurate measurements of the positions of the bright lines which many of the spectra exhibit; and much in this field of observation, therefore, remains to be accomplished. I purpose, from time to time as I shall have leisure, to make a series of observations, whose object shall be the actual numerical determination of the positions of the bright lines in the spectra of flames; and I have commenced the series with an examination of the spectra of the flames of compounds of carbon and hydrogen. In an investigation into the phenomena of flames, the compounds of carbon and hydrogen claim our first attention, as constituting the most important means of artificial illumination; for it is scarcely necessary to remark, that, with the grand exception of sun-light, the combustion of these substances is the source of nearly all the light and heat from which we derive such extensive benefits in the arts and in domestic economy. It will be found, moreover, that the spectra of carbohydrogen flames possess, in common, remarkable features, which seem as yet to have received little attention, but which promise to be of service in explaining the general phenomena of artificial light.

The history of the organic bases has now become as ardent an object of study among chemists as that of the as yet far more numerous group of acids; and it is generally admitted that the information obtained has been no less important and interesting. Of all organic alkaloids, those of opium and the cinchona barks doubtless occupy the first rank, whether we regard their value as remedial agents, or the remarkable facts which have been ascertained connected with their atomic relations, and the influence exerted by the latter, upon the theory of the science.

Having, several years ago, in the course of a prolonged investigation of the geological structure of the Appalachian chain of the United States, conducted partly in co-operation with my brother, Professor W. B, Rogers, as a purely scientific inquiry, partly by myself, in connection with a Government Survey of the State of Pennsylvania, discovered what we deemed important laws, applicable generally to all corrugated tracts of strata; and being prepared, by observations since made in the United States and in Europe, to extend their application, and give them a more general expression, I have thought that I could not select a more suitable subject for my first communication to the Royal Society of Edinburgh, than this portion of descriptive and dynamic geology, which has engaged much of my attention, theoretically and practically, for these many years. In presenting an outline of the views already arrived at, and published by us as a necessary part of the further generalizations since reached, I will refrain from repeating, in historical detail, what we have already written, but will give our conclusions in the form and with the brevity most compatible with clearness, referring to the printed papers and communications where the special topics included in this more general summary may be seen.

The experiments on the rain-fall in the lake district valleys were commenced in 1844,—the mountain gauges were planted in the spring of 1846,—the hygrometrical observations were instituted at the beginning of the year 1852,—and the entire investigation was brought to a close at the end of the year 1853, when its main objects were considered to have been satisfactorily answered and attained.

The tabular observations and general results of the inquiry up to the close of the year 1850, were communicated to the Royal Society of London in four papers, which are published in the Philosophical Transactions of that body. The fifth and concluding paper (comprising the observations, &c. for the years 1851—53 inclusive) I have now the honour to present to the Royal Society of Edinburgh.

In the course of those researches on Colour-Blindness, which, at intervals, I have recently been engaged in prosecuting, I have encountered some phenomena connected with normal vision, which I am desirous to submit to the consideration of the Society. Those phenomena I have already in part detailed, in the account of the researches referred to, and I shall not, accordingly, repeat the description of them here, to a greater extent than is essential to rendering intelligible the question which I wish to submit for discussion.

I venture to assume, that without adducing a lengthened series of authorities, I may take for granted, that, on the received theory of vision, the eye of man, as well as that of most of the lower animals, is regarded as essentially realizing, during the performance of its function of sight, the condition of a darkened chamber, or camera obscura. In more precise words: the theory in question teaches, that those rays of light, which reach the eye from the objects which they render visible, and entering at its front traverse all its transparent humours and membranes, last of all pierce the retina, and after making that impression upon it which is supposed to be the most important physical element of vision, are stopped, or absorbed by the dark pigment lining the choroid coat, and suffer extinction as visible rays. The dark surface of the choroid is thus held to abolish all the light which reaches it, so that none of the luminous rays return through the retina, or retrace their course across the chamber of the eye.

Preliminary §§ 97–101. Fundamental Principles of General Thermo-dynamics recapitulated.

97. Mechanical action may be derived from heat, and heat may be generated by mechanical action, by means of forces either acting between contiguous parts of bodies, or due to electric excitation; but in no other way known, or even conceivable, in the present state of science. Hence Thermo-dynamics falls naturally into two Divisions, of which the subjects are respectively, the relation of heat to the forces acting between contiguous parts of bodies, and the relation of heat to electrical agency. The investigations of the conditions under which thermodynamic effects are produced, in operations of any fluid or fluids, whether gaseous or liquid, or passing from one state to the other, or to or from the solid state, and the establishment of universal relations between the physical properties of all substances in these different states, which have been given in Parts I.-V. of the present series of papers, belong to that first great Division of Thermo-dynamics—to be completed (as is intended for future communications to the Royal Society) by the extension of similar researches to the thermo-elastic properties of solids.

1. A compass needle, although freely suspended, will not always point with accuracy in the direction of the magnetic meridian, for its magnetic axis may not be strictly parallel to its axis of figure; and hence, when a rigidly exact value of the magnetic declination is required, it is necessary to take the mean of at least two observations of the needle, first, in its usual position, and next, inverted. Some time ago it occurred to me that the declination obtained in the manner now described, will only be approximately correct, unless the inversion is accomplished with perfect accuracy; and I wished to ascertain the greatest residual error which, in given circumstances, is likely to affect the mean of two observations of a magnet, first in an erect, and then in an inverted position. I failed, however, to find any allusion to such errors in those works on magnetism to which I had access; and I was therefore obliged to investigate the subject for myself.

The investigation of which I am now about to give an account, was undertaken with the view of determining whether the structure of the Torbanehill mineral was similar to or unlike that of coal. I was aware that the subject would be brought before a court of law, and that many scientific persons of great eminence had already spent much time in the inquiry. With the understanding, therefore, that my evidence, should it be required, was to be limited to the structure of coal and of the mineral in question, I gave directions to Mr Bryson, the optician, of this city, to make thin sections of attested specimens of various coals and of the mineral, conceiving that a careful examination of them would easily determine the point. It was soon apparent, however, that a far more extended series of researches was necessary than I at first anticipated; but as it was also evident, from the marked structural differences which were observed in the sections, that the investigation would not be destitute of positive results, I determined on pursuing it to a conclusion.