comparatively small number of observations, it seems difficult to decide whether the apparently greater brilliancy towards the edge along the equatorial diameter is real, or is due to errors of observation. Column 7 gives the mean of all the measurements taken, and column 8 the probable error of this mean. If the sun had no atmosphere, its disk as seen from a distance would appear uniformly bright, since the light emitted by one square metre in any given direction is inversely as the cosine of the angle of emission, while, owing to foreshortening, its apparent area is proportional to the cosine of the same angle. Let us next suppose it surrounded by a homogeneous atmosphere not perfectly transparent. Evidently the absorption will depend on the distance which the light has to pass through it, and will be greatest at the edges, and least at the centre; or the disk will appear brightest at the centre and darkest at the exterior, as is actually the case. To determine the law of this variation, let the radius of the disk equal unity, the apparent distance of any point from the centre, h the height of the atmosphere, b the brightness of any portion of the disk were there no atmosphere, a the proportion of light which would traverse a thickness of the atmosphere equal to unity, or to the sun's radius. Call v also the distance the light from the point x must traverse before emerging from the solar atmosphere, and y the apparent brightness of the same point. It is readily proved = √(1 + 4) 2 — x2 -1-22, and that y=ba"; therefore, is the equation which gives the brightness of any point of the sun's disk, assuming that its atmosphere is homogeneous. From any three corresponding values of x and y we can compute a, b, and h. Assuming from the above observations y=1 for x = 0, y= .782 for x= .75, and y=.374 for x=1, and taking logarithms, we deduce the three equations of conditions: 0= log b h log a; = -.1068 log b + (√(1 + h)2 — .5625,6614) log a; Subtracting these equations, we eliminate b; and dividing one of the resultant equations by the other, eliminates a. We thus deduce the equation: √.4375 +2h + h2 — .25 √2h + h2 —.75h.6614 0. = To solve this equation, its first member was placed equal to m, and various values of h substituted; a curve was then constructed with m and h as coördinates, and a few trials readily gave the value of h corresponding to m=0. This affords an easy means of solving many equations not readily treated by the usual methods. The value of h thus found was a little less than unity. Substituting h=1 gives log α= .5835, a=.2609, log b .5835, and b=3.833. Or, if the effect of the solar atmosphere resembles that of a homogeneous atmosphere, its height must equal the radius of the sun, and its opacity be such that the light in the centre is only .26 of what it would be were the atmosphere removed; or the sun's brightness in the latter case would be throughout 3.8 times its present brightness at the centre. Substituting these values of a, b, and h in our first equation, gives logy.5835.5835 (√4—x2-√1 — x2), in which, by substituting various values of x, we deduce the corresponding values of y, the light at various points of the sun's disk. In Table I., the column headed Theor. gives the amount of light computed by this formula, and the last column the differences from the mean observations, M. Three other theoretical values were computed for these points, but those given in the table were retained as agreeing most nearly with observation. From these it appeared that a considerable variation in h did not alter the amount of light very materially, that a diminutive change of h of one-tenth increased the light between x=.6 and x.9 only half a per cent, and for other values of x altered y still less. Moreover, the differences in the last column of the table are evidently too regular to be due to accidental error, but rather show a real variation from theory, due to the fact that the atmosphere is not really homogeneous. We might assume that the law of the density is the same as that of the earth's atmosphere, or that, the height being taken in arithmetical progression, the densities will vary geometrically. But this leads to an equation which cannot be integrated, and, moreover, cannot be correct in fact, since it assumes that the temperature is uniform throughout. The great heat near the surface, by expanding the atmosphere in contact with it, diminishes its density, thus rendering it more nearly homogeneous than the above law would require; this effect is, however, counteracted by the tendency of the heavier gases to descend. It is a matter of interest to know not merely how much light is cut off by the atmosphere at the centre of the sun's disk, but also how much the whole light of the sun will be reduced by the same cause. Suppose the curve constructed with coördinates equal to x and y of the preceding table, and that a solid of revolution is generated by revolving it around the axis of Y: evidently, the volume of this solid will represent the total amount of light received by the observer from the whole of the sun's disk, and the volume of the circumscribing cylinder will equal that which would be received if the disk throughout had the same brightness as at the centre. The ratio of these two quantities is, however, obtainable by Simpson's formula, and gives the result 82.6, or the light is about five-sixths of what it would be if the disk had the same brightness at the edges as at the centre. Now, as shown above, the light at the centre is reduced by the atmosphere to 26.1 per cent. Hence the total reduction of the whole surface is .261 .826.216. And, since the light is reduced in every direction by the same amount, we may say that the sun would give out 4.64 times as much light if its atmosphere were removed. The results of this paper may therefore be summed up as follows. The light of the various parts of the sun's disk is measured by the modification of the Bunsen photometer here employed, and given in the accompanying table, with a probable error not exceeding one per cent except close to the edge. The light at the edge is about .4 of that at the centre. The variations in brightness are nearly those which would be produced by a homogeneous atmosphere of height equal to the sun's radius, and opacity such that only 26 per cent of the light is transmitted. There appears to be a slightly different distribution of the light along the polar, from that along the equatorial, diameter. If the atmosphere were removed, the brightness of the sun's disk would be uniform, and 3.83 times that of the centre of the disk at present. Moreover, the total amount of light would be increased 4.64 times. VI. TESTS OF A MAGNETO-ELECTRIC MACHINE. BY E. C. PICKERING AND D. P. STRANGE. THE rapidly increasing use of magneto-electric machines as a source of electricity renders accurate tests of the comparative advantages of the various forms and exact measurements of the currents gencrated under varying conditions very desirable. The machine employed in the following experiments was made by Mr. M. G. Farmer, and consists of a large electro-magnet wound with four coils soldered together at the ends, like four battery cells connected for quantity. Between the poles of this magnet a Siemens' armature is revolved, and both magnet and armature are included in the main circuit. The instrument is therefore extremely simple, and, when the circuit is broken, requires no power to run it except to overcome the friction of the bearings. The total weight is about 700 lbs., and the dimensions 33.5 by 21.5 inches (85 × 55 cms.), with a height of 14 inches (37 cms.). To avoid heating, a water space is left close to the armature, but this is required only when the resistance of the circuit is small. The quantities to be measured were as follows: 1st, velocity of rotation of armature; 2d, power required; 3d, strength of current with various speeds and resistances; 4th, electro-motive force under the same conditions; 5th, when the current is used to produce a light, a measure of the latter is candle-power, Power. The boilers and engine of the Institute were used as a source of power. The nominal capacity of the boilers was sixty-six, and of the engine fifteen horse-power; but, owing to various difficulties beyond the control of the writers, only a small portion of this was available, and that only for limited periods of time. A belt passed from the fly-wheel of the engine over a countershaft in the Physical Laboratory, giving it a velocity of about 500 turns per minute. A set of five cone pulleys were attached, by which a speed of 333, 410, 500, 610, and 750 turns could, by shifting a belt, be given to a second shaft. The latter carried a wheel 20 inches in diameter, and drove the machine by a belt passing over a pulley 8 inches in diameter attached to the armature. As the speed of the engine varied somewhat, a speed of from 800 to 2100 turns per minute was thus obtained. Various plans were tried to measure the power employed. For the earlier experiments a Batchelder dynamometer was used, in which the motion was transmitted through four bevel-gears, and the moment of tension measured by a spring-balance and weights. The instrument was not however intended to be run at such high speeds, and the gears were very noisy. Speed. The number of revolutions per minute is so important a factor in these measurements that it must be constantly determined. At first, a common shaft speeder was employed; but, apart from its want of accuracy, its constant use was laborious, and it showed only the total number of turns during a minute, and not the speed at any intermediate instant. A device was accordingly employed, constructed by Mr. J. B. Henck, Jr., by which these difficulties are completely avoided. The plan is not new, having been published in a modified VOL. X. (N. S. 11.) 28 form many years ago in Nicholson's Mechanics and elsewhere. Three vertical gas-pipes are placed side by side, and connected together below; then half filled with mercury, and so mounted that they may revolve around the axis of the central pipe; a glass tube filled with water is attached to the latter, and serves to show the position of the mercury. Motion was transmitted to the whole from the horizontal shaft of the machine by a spiral spring, as in a dental lathe, but afterwards this was replaced by a pair of bevel-gears. If now the machine is set in motion, the mercury is by centrifugal force thrown from the central to the outer tubes, and the water in the glass tube falls. A graduated scale shows the position of the water, which remains very constant as long as the velocity is uniform, and by its motion shows the slightest variation in speed. The reduction is effected by noting the water level with various velocities as measured by a shaft-speeder, and constructing a curve with coördinates equal to these two quantities. If the tubes are exactly parallel and of uniform diameter, this curve will be a parabola, with axis vertical and parameter determined by the equation y=473 × 10-n'd2, in which d is the distance of the outer tubes from the centre in inches, and n the number of turns per second. Evidently an inch would correspond to a much greater change in velocity at high than at low speeds, and accordingly the open ends of the outer tubes were bent in towards the centre. This had the additional advantage of preventing the mercury from being thrown out, and of greatly increasing the range of the instrument. As actually constructed, the speed in turns per minute very nearly equalled the square of the depression of the water level in tenths of an inch. Resistances. A difficulty at once presented itself in varying the resistance of the circuit, since resistance coils of the ordinary form would be at once injured or even melted by the immense quantity of electricity transmitted. Accordingly a set of resistances were prepared by stretching some uncovered German silver wire along the wall of the laboratory, so as to form nine loops, of eighty feet each, of No. 28 wire. As the diameter is .017 inches, the surface exposed to radiation is about 460 square inches; and, as the air circulates freely around them, there is no difficulty from their heating, even when the machine is connected directly with their terminals. Each of these loops has a resistance of 36.9 ohms, and one or more may be thrown into circuit by a switch. For smaller resistances, a similar device was employed. A frame, 3 feet wide and 6 feet high, was covered on both sides with horizontal wires, passing around screws so as to form 30 loops of No. 22 wire (diameter .029") and 55 loops of No. 16 wire |