Now I have been able to establish the following theorem of derivation as a particular case of a more general one of which the clue is in my hands: [1, λ, μ, v, . . .] = [λ, μ, v, . . .] +Σ(λ2—λ)[1, λ—2, μ, v, . . .] Suppose now that λ, μ, v,... all become unity, and that we call [1, 1, 1,... to n terms]=n, then the theorem above stated gives the relation Qn=Qn−1+(n−1) (n—2) Nn—2• But by virtue of the form of the equations for finding fx, I know independently that , is the product of n terms of the progression n 1, 1, 2, 2, 3, 3, 4, ... Hence we have one particular solution of the above equation in differences. To find the second, if we make 2, and 2, 1 and 2 respectively instead of 1, 1, it will be found that the nth term becomes the product of n terms of the analogous progression 1,2,2,4,4,6, 6.... Thus, then, we are in possession of the complete integral of the equation viz. = Ux+1=Ux+(x2—x) ux-15 U2 C.12.32.52... (2x-1)2 + K22.42...(2x-2)22x, U2x+1=C.12.32.5 ... (2x-1)2 (2x + 1) + K.22. 42... (2x)2. Writing u=1.2.3... (x-1)v, the above equation takes the remarkably simple form of any two of the quantities i, j, k, l happens to become equal to the sum of the other two, the order sinks and is either 8 or 7; I am not quite cer tain which at present, although it is more probably the former. Whether taken under this or the original form, the equation will be found to lie outside the cases of integrable linear difference of equations of the second order with linear or quadratic coefficients given by the late Mr. Boole in his valuable treatise on finite differences. In the second form the solution ought by Laplace's method to be representable by a definite integral. Expressed under the more ordinary form the integral is as follows: 3.5.7...(2x-1) 2.4.6... (2x) 2.4. 6 (2x-2) 1.3.5...(2x-1)' V2x =C ... +k V2x-1=C3.5.7....(2x-1) +h 2.4.6...(2x-2) 1.3.5...(2x-3)' The romance of algebra presents few episodes more wonderful than this specimen of the way in which the determination of the degree of an equation resulting from elimination can be made to contribute a new and by no means obvious fact to the Calculus of Differences. Athenæum Club, XXXI. Proceedings of Learned Societies. ROYAL SOCIETY. [Continued from p. 145.] December 17, 1868.-Captain Richards, R.N., Vice-President, in the Chair. HE following communication was read :— THE "On the Measurement of the Luminous Intensity of Light.” By William Crookes, F.R.S. &c. The measurement of the luminous intensity of a ray of light is a problem the solution of which has been repeatedly attempted, but with less satisfactory results than the endeavours to measure the other radiant forces. The problem is susceptible of two divisions, the absolute and the relative measurement of light. A relative photometer is one in which the observer has only to ascertain the relative illuminating powers of two sources of light, one of which is kept as uniform as possible, the other being the light whose intensity is to be determined. It is therefore evident that one great thing to be aimed at is an absolutely uniform source of light. In the ordinary process of photometry the standard used is a candle, defined by Act of Parliament as a sperm of six to the pound, burning at the rate of 120 grains per hour." This, however, is found to be very variable, and many observers have altogether condemned the employment of test-candles as light-measures. The author has taken some pains to devise a source of light which should be at the same time fairly uniform in its results, would not vary by keeping, and would be capable of accurate imitation at any time and in any part of the world by mere description. The absence of these conditions seems to be one of the greatest objections to the sperm-candle. It would be impossible for an observer on the continent, ten or twenty years hence, from a written description of the sperm-candle now in use, to make a standard which would bring his photometric results into relation with those obtained here. Without presuming to say that he has satisfactorily solved all difficulties, the writer believes that he has advanced some distance in the right direction, and pointed out the road for further improvement. A glass lamp is taken of about 2 ounces capacity, the aperture in the neck being 0.25 inch in diameter; another aperture at the side allows the liquid fuel to be introduced; this consists of alcohol of sp. gr. 0.805, and pure benzol boiling at 81° C., which are mixed together in the proportion of five volumes of the former and one of the latter. The wick-holder consists of a platinum tube, and the wick is made of fifty-two pieces of platinum wire, each 0.01 inch in diameter. The flame of this lamp forms a perfectly shaped cone, the extremity being sharp, and having no tendency to smoke; without flicker or movements of any kind, it burns when protected from currents of air at a uniform rate of 136 grains per hour. There is no doubt that this flame is very much more uniform than that of the sperm-candle sold for photometric purposes. Tested against a candle, considerable variations in relative illuminating power have been observed; but on placing two of these lamps in opposition, no such variations have been detected. The instrument devised for measuring the relative intensities of the standard and other lights is next described; it has this in common with that of Arago described in 1833, as well as with those described in 1853 by Bernard, and in 1854 by Babinet, that the phenomena of polarized light are used for effecting the desired end*. But it is believed that the present arrangement is quite new, and it certainly appears to answer the purpose in a way which leaves little to be desired. The instrument cannot be described without the aid of drawings, which accompany the original paper; but its mode of action may be understood by the following description. The standard lamp being placed on one of the supporting pillars which slide along a graduated stem, it is moved along the bar to a convenient distance, depending on the intensity of the light to be measured. The light to be compared is then fixed in a similar way on the other side of the instrument. On looking through the eyepiece two brightly luminous disks will be seen, of different colours. One of the lights must now be slid along the scale until the two disks of light, as seen in the eyepiece, are equal in tint. Equality of illumination is easily obtained; for, as the eye is observing two adjacent disks of light which pass rapidly from red-green to green-red, through a neutral point of no colour, there is no difficulty in hitting this point with great precision. Squaring the distance between the flames and the centre will give inversely their relative intensities. The delicacy of this instrument is very great. With two lamps, each about 24 inches from the centre, it is easy to distinguish a movement of one of them to the extent of one-tenth of an inch to or fro; and by using the polarimeter an accuracy exceeding this can be attained. The employment of a photometer of this kind enables us to compare lights of different colours with one another. So long as the observer, by the eyepiece alone, has to compare the relative intensities of two surfaces respectively illuminated by the lights under trial, it is evident that, unless they are of the same tint, it is impos Since writing the above, I have ascertained that M. Jamin had previously devised a photometer in which the principle adopted in the one here described is employed, although it is carried out in a different and, as I believe, a less perfect manner.-W. C., Dec. 16, 1868. sible to obtain that absolute equality of illumination in the instrument which is requisite for a comparison. By the unaided eye one cannot tell which is the brighter half of a paper disk illuminated on one side with a reddish, and on the other with a yellowish light; but by using the photometer here described the problem becomes practicable. When the contrasts of colour are very strong (when, for instance, one is a bright green and the other scarlet) there is difficulty in estimating the exact point of neutrality; but this only diminishes the accuracy of the comparison, and does not render it impossible, as it would be according to other systems. January 7, 1869.-Lieut.-General Sabine, President, in the Chair. The following communication was read: "On the Mechanical Possibility of the Descent of Glaciers by their Weight only." By the Rev. Henry Moseley, M.A., Canon of Bristol, F.R.S., Instit. Imp. Sc. Paris, Corresp. All the parts of a glacier do not descend with a common motion; it moves faster at its surface than deeper down, and at the centre of its surface than at its edges. It does not only come down bodily, but with different motions of its different parts; so that if a transverse section were made through it, the ice would be found to be moving differently at every point of that section. This fact*, which appears first to have been made known by M. Rendu, Bishop of Annecy, has since been confirmed by the measurements of Agassiz, Forbes, and Tyndall. There is a constant displacement of the particles of the ice over one another, and alongside one another, to which is opposed that force of resistance which is known in mechanics as shearing force. By the property of ice called regelation, when any surface of ice so sheared is brought into contact with another similar surface, it unites with it, so as to form, of the two, one continuous mass. Thus a slow displacement of shearing, by which different similar surfaces were continually being brought into the presence and contact of one another, would exhibit all the phenomena of the motion of glacier ice. Between this resistance to shearing and the force, whatever it may be, which tends to bring the glacier down, there must be a mechanical relation, so that if the shearing resistance were greater the force would be insufficient to cause the descent. The shearing *The remains of the guides lost in 1820, in Dr. Hamel's attempt to ascend Mont Blanc, were found imbedded in the ice of the Glacier des Bossons in 1863. "The men and their things were torn to pieces, and widely separated by many feet. All around them the ice was covered in every direction for twenty or thirty feet with the hair of one knapsack, spread over an area of three or four hundred times greater than that of the knapsack." "This," says Mr. Cowell, from whose paper read before the Alpine Club in April 1864 the above quotation is made, "is not an isolated example of the scattering that takes place in or on a glacier, for I myself saw on the Theodule Glacier the remains of the Syndic of Val Tournanche scattered over a space of several acres." force of cast iron, for instance, is so great that, although its weight is also very great, it is highly improbable a mass of cast iron would descend if it were made to fill the channel of the Mer de Glace, as the glacier does, because its weight would be found insufficient to overcome its resistance to shearing, and thus to supply the work necessary to those internal displacements, of which a glacier is the subject, or even to shear over the irregularities of the rocky channel. The same is probably true of any other metal. I can find no discussion which has for its object to determine this mechanical relation between what is assumed to be the cause of the descent of a glacier, and the effect produced-to show that the work of its weight (supposing that alone to cause it to descend) is equal to the works of the several resistances, internal and external, which are actually overcome in its descent. It is my object to establish such a relation. The forces which oppose themselves to the descent of a glacier are:-1st. The resistance to the sliding motion of one part of a piece of solid ice on the surface of another, which is taking place continually throughout the mass of the glacier, by reason of the different velocities with which its different parts move. This kind of resistance will be called in this paper (for shortness) shear, the unit of shear being the pressure in lbs. necessary to overcome the resistance to shearing of one square inch, which may be presumed to be constant throughout the mass of the glacier. 2ndly. The friction of the superimposed laminæ of the glacier (which move with different velocities) on one another, which is greater in the lower ones than the upper. 3rdly. The resistance to abrasion, or shearing of the ice, at the bottom of the glacier, and on the sides of its channel, caused by the roughnesses of the rock, the projections of which insert themselves into its mass, and into the cavities of which it moulds itself. 4thly. The friction of the ice in contact with the bottom and sides so sheared over or abraded. If the whole mechanical work of these several resistances in a glacier could be determined, as it regards its descent, for any relatively small time, one day for instance, and also the work of its weight in favour of its descent during that day, then, by the principle of "virtual velocities" (supposing the glacier to descend by its weight only), the aggregate of the work of these resistances, opposed to its descent, would be equal to the work of its weight, in favour of it. It is, of course, impossible to represent this equality mathematically, in respect to a glacier having a variable direction and an irregular channel and slope; but in respect to an imaginary one, having a constant direction and a uniform channel and slope, it is possible. Let such a glacier be imagined, of unlimited length, lying on an even slope, and having a uniform rectangular channel, to which it fits accurately, and which is of a uniform roughness sufficient to tear off the surface of the glacier as it advances. Such a glacier would |