of the star, or the disks will touch each other. They correspond to the first and last contacts of an eclipse. The orbit is projected into an ellipse whose major axis, a, equals the true distance of the centres, and whose minor axis, b, equals the distance at the time of greatest obscuration. When r 0.746, b = 10.7460.254. For other values of r, b must be determined from a computation of the area eclipsed, by successive approximations, until such a value is found as will reduce the light to 0.416. If x and y are the co-ordinates of the point in the orbit reached by the satellite at the time of first contact, by the properties of the ellipse x = a sin w, and y = b cos w. The square of the distance of the centres, or D2, may be written D2 = (1 + r)2 = (x2 + y2) = a2 sin2 w + b2 cos2 w Since w 23°.0, (1+r)2= 0.153 a2+0.847 62. Substituting the proper values of r and b, a may be deduced. The cosine of the inclination, i, of the orbit will equal. The three lines a of Table IX. give the values of a, b, and i computed by these formulas for the minimum value of r = 0.764, for r = 1.000, and for r = 2.000. There is no maximum value of r, which may be indefinitely large. Let R be any large value of r, and let a = R+ A, b=R+B, and D=R+d; substituting these values in the formula, D2 = a2 sin2 w + b2 cos2 w, the terms containing R2 cancel each other, and we have 2 R d= 2 RA sin2 w + 2 RB cos2 w, omitting the terms not containing R, since when R is very large they may be neglected. Dividing both sides by 2 R gives d = A sin2 w + B cos2 w. When w=23°, d must equal 1, and when w = 0°, B will equal −0.132, since the arc of the large circle becomes sensibly a straight line, and the segment whose versed sine is 1.000 — 0.132 has an area of 0.416, or the minimum area of the uneclipsed portion. From these values, we may deduce A 7.300. The two axes, therefore, become R0.132 and R+7.300. The inclination in this case continually diminishes as R increases, and would equal zero if R became infinite. The residuals which will be deduced below at first led to the belief that the phenomenon might be that of an annular eclipse. This case has therefore been included to show the change effected in the variation of the light, although the residuals are not materially reduced. If the eclipse is annular, the value of r must be 0.764. The value of b cannot be determined directly, but must be deduced from the times of internal and external contact. The interval between the internal contacts is assumed to be 24 minutes, or that during which the satellite moves through 2° of longitude. In the equation D2 = a2 sin2 w + b2 cos2 w, we have for w = 1o, D= (1 − r) = 0.236, and as before for w= 23°, D=1+r=1.764. From these conditions the values of a and b given in the last column of Table IX. are deduced. We must next compute the amount of obscuration at the end of each half-hour, for the various values of r. The distance between the centres is first computed by the equation Da2 - (a2b2) cos2 w, substituting successively, w 2°.5, 5°.0, 7°.5, 10.°0, 12°.6, 15°.1, 17°.6, and 20°.1. The first part of Table X. gives the values of D corresponding to those assigned to r at the head of each column. The triangles formed by the centres of the two bodies and one end of the segment now become known, since their three sides equal 1, r, and D. Calling the angle at the centre of the luminous body a, we have r2 = 12 + D2 - 2 D cos a. From this we deduce cos a and versin a, or the height of the segment bounded by the circle having a radius unity. The height of the other segment will equal RD + cos a, from which the areas of the segment, and consequently of the uneclipsed portion, may be deduced. This area is given in the second portion. of the table. For comparison the observed light is repeated in the last column from the last column of Table VIII. The residuals, or the observed values minus those computed with each value of r, are given in the third part of Table X. The residuals are all zero when the time equals 0.0 or 4.6, and are therefore omitted. The average residuals are given in the last line. The residuals are all expressed in terms of the full light of the star. They therefore represent a larger error expressed in logarithms, or stellar magnitudes, when the star is faint than when it is bright. If reduced to logarithms their mean values become .008, .012, .027, .049, .008. Dividing these quantities by 0.4 to reduce them to magnitudes, we see that while a large value of r would give an average residual of over one tenth of a magnitude, the value of r = 0.764 would make this quantity less than two hundredths of a magnitude. In all of them, however, there is a distinct systematic variation, the computed light being too small when t is large, and sometimes becoming too large when t is small. It appeared that this error might be reduced by assuming that the eclipse was annular, or that the light retained its minimum value for a short time. The corresponding residuals are given in the last column. They reduce the positive residuals when the star is faint, but do not sensibly affect the others, although the time between the internal contacts is assumed to be twenty-four minutes. The observations scarcely admit so great an interval, and certainly would not justify its increase. As the average residual is not diminished by the assumption of an annular eclipse, and as the observations do not indicate that the light remains constant during the minimum, we cannot do better than to assume the value of r = 0.764, and adopt the values of the second column of the table. Several explanations may be offered of the small systematic error that remains. The most plausible seems to be that derived from the residuals given in the last column of Table VII. They show that, from a comparison of the estimated grades of Schönfeld with the measures of Wolff, that Schönfeld estimated the light too faint when the star was faint, and too bright when the star was bright. In other words, that a grade did not have the same values when expressed in logarithms for a faint as for a bright star. Assuming the photometric measures of Wolff to be free from systematic error, we should therefore increase the estimates of Schönfeld when the star was faint, and diminish them when it was bright, without affecting the actual maximum and minimum values. Such a correction would make the systematic error noted above disappear, or even give it an opposite sign. This view receives a slight confirmation from the measures of Seidel, but the accidental discrepancies far exceed this small systematic error. We may therefore conclude that the computed light agrees with observation as closely as the brightness of the fundamental stars is at present known, and there is no evidence of a real systematic difference between the two. Another explanation of the residuals in Table X. has suggested itself. The presence of lines in stellar spectra leads to the belief that the stars, like our Sun, are surrounded by an absorbing atmosphere. They also, therefore, probably resemble it in presenting a disk brighter in the centre than at the edges, owing to the greater thickness of the atmosphere and consequent greater absorption at the edges. The effect of such an absorption is best determined by the consideration that if, owing to absorption, the average light of the eclipsed portion is less than that of the whole disk, the effect of the atmosphere will be to diminish the proportion of the light cut off; in the opposite case, it will increase it. Now when a small portion only of the star is eclipsed, evidently the average light of this portion, since it lies near the edge, must be less than that of the whole. The atmosphere, although then diminishing the light of the remaining portion, will not reduce it as much as it does that of the entire disk; the relative light will therefore be increased. On the other hand, when a large part of the eclipsed portion is from the central and brightest portion the opposite effect will be produced. We should therefore expect, when t is large, that the computed light should be increased. When t is small, it may be diminished. In the case of the Sun the effect is so slight, except close to the borders, that the previous explanation seems more probable. We return now to the consideration of differences in the rate of diminution and increase of the light. The observations ought to give this quantity with much accuracy. An error in estimating the light of the standard stars will not sensibly affect it, since the same stars are used in measuring the increase and diminution. The effect of atmospheric absorption is reduced, since some of the comparison stars are always above and others below the variable, and besides, although, when observed before passing the meridian, the star is brighter when increasing than when diminishing, yet the opposite effect is produced when the star is west of the meridian. Nevertheless this difference is doubted by many astronomers, and if it exists it is evident that an important correction should be applied to the observed minima of Algol. If the curve found by Schönfeld is correct, an error of ten minutes in the time of the minimum might be caused by comparing with a star like ɛ Persei, having a brightness of about twelve grades, and taking the mean of the times when the two stars appeared equal. Three explanations may be offered for this phenomena. First, that the satellite is not spherical, but egg-shaped, and that the large end is turned forwards; or that the satellite is of unequal density, and that the heaviest portion is forward. In this case the centre of gravity of the disk would follow that of the satellite, or for a given distance of the centres the interposed area would be greater when the satellite was passing off, than when coming on. So great a deviation from the spherical shape would be needed to produce the observed difference |