UC-NRLF 
 
 3M 437 
 
' 
 
 A STUDY 
 
 OF 
 
 THE LIGHT CURVE OF THE VARIABLE STAR U PEGASI 
 
 BASED ON 
 
 THE OBSERVATIONS OF HARVARD COLLEGE OBSERVATORY CIRCULAR NO. 23. 
 
 BULLETIN NO. I. 
 
 OP 
 
 THE ASTRONOMICAL OBSERVATORY 
 
 OF THE 
 
 UNIVERSITY OF ILLINOIS, 
 
 UKBANA, ILLINOIS. 
 
 BY 
 
 G. W. I MYERS, DIRECTOR. 
 
 CAMBRIDGE, U.S.A.: 
 PRINTED FOR THE UNIVERSITY OF ILLINOIS, 
 
 at tljc JSmbrvsitB 
 1898. 
 
ASTRONOMY LIBRARY 
 
A STUDY OF THE LIGHT CURVE 
 
 OP 
 
 THE VARIABLE STAR U PEGASI. 
 
 PROFESSOR PICKERING has shown in Harvard College Observatory Circular No. 23, that 
 U Pegasi no longer deserves the distinction of being considered the variable of shortest known 
 period. Contrary to the usual form of contestant, in the present instance, the disputant for 
 pre-eminence in this particular is not a newly discovered variable of shorter period than any 
 hitherto known, but is the variable a> Centauri 19, discovered by Baily some time since and found 
 to have the period 7 A ll m . Manifestly, therefore, U Pegasi, whose period has until recently been 
 regarded as lying between 3\0 and 5 h .6, has been turned down the list, not because of the exces- 
 sive shortness of the period of some other star. The reason for the change lies in the fact that 
 the inequality of brightness of the alternate minima of U Pegasi escaped detection, until 
 Professor Pickering's discussion revealed it last winter. His observations, published in the form 
 of a light curve and reproduced in substance in Plate I. accompanying this paper, showed 
 the most probable period based upon all preceding observations to be about 4 A .5 ; but that, in 
 view of the failure of former observers to recognize the difference of brightness of the minima, 
 this period should be doubled. Applying a slight correction to the double value, shown to be 
 justified by more recent observations, he states, as the best value for the period-length of this 
 star 8* 59 m 41*. The mean value of the brightness at the two approximately equal maxima 
 is 9.30 ; at the secondary minimum, the brightness is 9 m .75, and at the primary it is 9 m .90. 
 The plate referred to gives the observations on such a scale that one division in the ordinates 
 corresponds to 0.1 magnitude and, in the abscissas, to half an hour. The above mentioned 
 circular states that the total number of settings here represented is 2784 and that the time of 
 observation, including rests, is 30 hours. Each dot in the plate represents 80 settings, the dots 
 being formed by the method of overlapping means. 
 
 The least difference of stellar brightness of whose existence the eye can be certain, being 
 about 0.1 of a magnitude, and the difference of brightness between the primary and secondary 
 minima, as stated in the Circular, lying so near this limit, i. e. = 0.15 of a magnitude, there would 
 seem to be just cause for suspicion that this apparent difference has arisen from the rather large 
 accidental errors always attaching to photometric observations. In view of the almost uniformly 
 high degree of excellence attained in the past by Professor Pickering's forms of photometer, 
 it cannot be denied that the results of photometric measures are on the whole to be ascribed a 
 far higher measure of accuracy than belongs to photometric estimates. A recent personal study 
 of /3 Lyrae's light variation made with one of Professor Pickering's polarization photometers 
 removes from the writer's mind the last vestige of doubt as to the certainty of the existence of 
 this difference of brightness at the minima. But whatever doubt may have existed for a time as 
 to its reality, it would seem that the following statements of Professor Pickering in the Ap. J. 
 for March of this year, ought to dispel it quite effectually. "Twelve observations, each consisting 
 of sixteen settings, were made when the star was within twenty minutes of its primary minimum. 
 
 ffi'753275 
 
A STUDY OF THE LIGHT CUUVE 
 
 Deriving from each of these, by means of the light curve, the magnitude of this minimum, 
 we obtain on Oct. 18, 1897, 9.89, 9.94, and 9.96; on Dec. 30, 9.90, 9.95, and 9.93; on 
 Jan. 1, 1898, 9.93, 9.86, and 9.85 ; on Jan. 5, 9.85, and on Jan. 7, 9.86 and 9.88. Mean of 
 all = 9.90 ; greatest value = 9.96 ; least value = 9.85 and average deviation = 0.035. Similarly, 
 fourteen observations were taken within twenty minutes of the secondary minimum with the 
 results on Oct. 18, 1897, 9.75 and 9.71 ; on Oct. 29, 9.74, 9.69, 9.70 and 9.70 ; on Dec. 28, 
 9.78, 9.77, 9.76 and 9.80; on Jan. 3, 1898, 9.77, 9.77, 9.74 and 9.78. Mean of all = 9.75 ; 
 greatest value = 9.80 ; least value = 9.69, and average deviation =. 0.029." The probable errors 
 would, of course, be smaller than the " average deviations." Obviously, average deviations, 
 probable errors, and the like, mean nothing at all here, or they mean that an error in the great- 
 est value of the primary minimum large enough to make it equal to even the least value at the 
 secondary cannot be entertained as a probability, since it would mean the commission of a 
 systematic error nearly twice as great as the average deviation and more than twice as great as 
 the probable error. The chances against this would be a little worse than 1 to 5.2. The inter- 
 nal evidence of the observations is, it would seem, quite conclusive in favor of the reality of the 
 discrepancy. The statements just quoted show, moreover, that especial attention was directed 
 to the point in question, and it seems therefore scarcely reasonable to suspect that, under such 
 circumstances, an error of 0.15 of a magnitude could elude certain detection and confirmation. 
 
 Assuming the reality of this difference, the light curve appears to be susceptible of treatment 
 by essentially the same method as that adapted and used by the writer in his recent discussion 
 of Beta Lyrae's light curve entitled : UNTERSUCHUNGEN DEBEB DEN LICHTWECHSEL DBS STERNES 
 /3 LYRAE, Muenchen, 1896. It is the purpose of this Bulletin to present the results and an out- 
 line of the method used in a recent study of U Pegasi, based essentially upon the observations 
 of Pickering's Circular No. 23, and by the method 'more fully developed in the foregoing disser- 
 tation. The fundamental hypothesis underlying the whole discussion is that the light curve of 
 U Pegasi is capable of being explained on the satellite theory. 
 
 ECCENTRICITY. 
 
 The uncertainty in the instants of maximum brightness as indicated by the light curve of 
 Plate I., obviously precludes the possibility of deriving an approximate value of the orbital 
 
 eccentricity of the component from the 
 chief epochs of light variation, as was done 
 with /3 Lyrae. One may readily convince 
 himself by considerations adduced below, 
 however, that this eccentricity must be 
 quite small. 
 
 Assuming the light fluctuations to be 
 due to the mutual eclipses of two unequally 
 bright bodies, we should have the chief 
 epochs occurring when the relative posi- 
 tions of the components are as indicated in 
 the subjoined figure. That the bodies are 
 unequally bright, follows at once from the 
 consideration that at Min. I. the brightness 
 of the star is reduced by 41 per cent of its 
 maximum brightness, and at Min. II. by 
 only 31 per cent ; unless the orbital eccen- 
 tricity is assumed quite large. It will now be shown that the latter cannot be the case. 
 Assuming also provisionally, that both bodies are spheres, a lower limit for the eclipse- 
 
 tVIHIMUM Jl. 
 
OF THE VARIABLE STAR U PEGASI. 3 
 
 duration at Min. I. can bo easily obtained from the observational curve given in Fig. 1. A little 
 reflection will make it clear tbat the shorter the eclipse-duration be taken, the larger will be the 
 corresponding distance between centres of the components. If, then, we assume that the eclipse 
 has not begun until the light curve has fallen quite appreciably and that it has ended shortly 
 before the curve ceases to rise, we shall obtain a value for the duration of the eclipse, at all 
 events short enough, perhaps too short, and the corresponding value of the distance of cen- 
 tres must be at all events great enough perhaps too great. Proceeding thus, I obtain 3*.3 for 
 the interval shorter than which the eclipse-duration at Min. I. cannot be. The corresponding 
 value of the distance between centres may then be regarded as fixing a superior limit for this 
 orbital element. 
 
 Calling the radius of the larger component unity and of the smaller , the radius vector of 
 the true orbit, r, one-half the distance between the nearest points of the positions of the com- 
 panions at the beginning and end of the eclipse, x, and for this roughly approximate purpose, 
 assuming e to be zero, we have from the figure : 
 
 CPC' > 3.3 fji = 132 (p = Zir/P = 360/9 = 40) 
 Hence, 
 
 CP D > 66 and r < (x + K ) esc 66 
 
 < 1.0946 (x + K ) 
 
 But since x ^ 1 and K < 1, we shall have r ^ 2.189 times the radius of the larger com- 
 panion. So small a distance of centres relative to the dimensions of the primary, coupled 
 with a large orbital eccentricity, would be highly improbable theoretically in any case, and 
 assuming distinct duplicity, would be a physical impossibility on any other hypothesis than 
 that the extent of the secondary is quite inconsiderable compared with that of the primary. 
 The approximately equal fall of brightness at the minima, together with the similarity of form 
 of the light curve in the neighborhood of these two chief epochs, argues strongly for the view 
 that the form and dimensions of the companions cannot be widely different, and this latter view 
 is still further supported by the fact that the relative brightness of the components is found 
 later, independently of any hypothesis regarding the ratio of the radii, to be about 0.8. 
 
 It may therefore be assumed as a first approximation that e = 0, and we shall now proceed 
 to determine the value of the ratio of the brightness of the companions and to fix the limits 
 within which the ratio of the radii must be comprised. We shall then undertake to find the 
 most probable value of this latter ratio by direct reference to the light curve of the star- 
 
 CIRCULAR ORBITAL ELEMENTS AND LIGHT RATIO OF THE COMPONENTS OF U PEGASI. 
 
 The chief epochs of the light curve shall be designated in order from left to right in Figure 1 
 as Min. I., Max. I., Min. II. and Max. II. From the curve Max. I. is seen to have a brightness 
 of 9.32 magnitude and Max. II. of 9.34 magnitude, so that the mean value 9 m .33 has been used 
 throughout the discussion for the brightness at both the maxima. For the brightness at Min. I., 
 the value 9.90 magnitude has been used and for Min. II., 9.75 magnitude. Reducing these 
 differences in stellar magnitudes at the chief epochs of variability to their equivalent light 
 ratios, by the aid of Pogson's scale, we obtain: 
 
 Brightness at Min. II. 
 
 Brightness at Min. I. 
 
 Brightness at Mean Max. 
 
 Brightness at Min. 1. 
 
A STUDY OF THE LIGHT CURVE 
 
 Ectaining the nomenclature of the foregoing paragraph, calling the light ratio of the com- 
 ponents X and the portion of the discs common to both bodies at the middle of the eclipses a, 
 the preceding equations give the following : 
 
 1 + 2 X - a K 2 X 
 (1) - = c 
 
 K 
 
 (2) - = m. 
 
 2 
 
 If it be thought desirable to include the possibility of a flattening of the discs, we may 
 assume, as a means of making a first approximation to the general effect of such deformation, 
 that the bodies are similar ellipsoids of revolution and designate by q, the common ratio of the 
 semi-major to the semi-minor axis, whereupon equation (2) must be replaced by 
 
 i j. if \ 
 
 /n \ ' K ** 
 
 (2a) q = m 
 
 (Conf. Ap. J. Vol. VII., p. 13, where a 2 should be stricken from the numerator of (e).) 
 From (1) and (2 a) we find readily 
 
 aK 2 X 
 
 and a K!! 
 
 whence, dividing, we get 
 
 (5) X = (m-cq)/(m-q). 
 
 Neglecting the flattening provisionally, i.e., putting q 1, (5) gives, when the foregoing 
 values of c and m are substituted, 
 
 X = 0.7865. 
 From (3) and (1), we obtain 
 
 m m co 
 
 . a- 
 
 K* m q m q 
 and (4) gives 
 
 a 
 m q = m 
 
 K 2 X 
 
 Since now, a K 2 and 1 + K 2 X are essentially positive, being quantities of light, this latter 
 relation shows that m must be greater than q. Consequently, 
 
 G) 
 
 m 
 
 da m q 
 
 is also a positive magnitude. (l//c 2 ) and a therefore, increase and decrease together, so that 
 the maximum value of a corresponds to the maximum value of (l/ 2 ). 
 If now, 
 
 K 2 ^ 1, then a < 1 (from geometrical considerations), 
 and it follows from (6) that, 
 
 , or K* > 
 
 K" m q c q 
 
 But if, 
 
 K" > 1, we may put a < (also for geometrical reasons), 
 
OF THE VARIABLE STAR U PEGASI. 
 
 We have from (6), 
 
 and hence, 
 
 m 
 
 m q 
 
 m cq 
 
 m cq 
 Summing up both contingencies into a single condition, there results : 
 
 (?) 
 
 From this, we have also, 
 and finally, 
 
 cq 
 
 m cq 
 
 cq 
 
 m cq 
 
 m 
 
 -*-- \d 
 
 Substituting now the former values of m and c, we obtain 
 
 q > 0.787. 
 
 It does not therefore appear to be necessary to assume the existence of a flattening for U 
 Pegasi, such as was shown to be necessary in my Dissertation on Beta Lyrae, p. 30, for the 
 latter star. 
 
 Taking again the value of q as unity, and substituting in (7) we find : 
 
 0.6014 < K S < 1.845, or 0.7755 < K < 1.358. 
 
 The following test values distributed linearly over this interval were, therefore, selected for 
 criteria to an approximation to : 
 
 0.80, 0.85, 1.00, 1.15 and 1.35, 
 
 and for each of these values a light curve was computed by the method and with the results 
 given below. 
 
 Using the portion of the light curve lying within 1.5 hours before and after Min. I., and the 
 notation (v, Fig. 3) and equations developed in my dissertation and published in the Ap. J. for 
 Jan., 1898, 1 have to compute the values of M and H from the data furnished by the Hght 
 curve and then for K. < 1, to solve the transcendental equations : 
 
 and for > 1, 
 
 M = <f> + K 2 <{>" - K sin (<" + 
 H = K? <t>' <f> + K sin (<#>' 
 
 M = </> + K 
 
 H = <f>! - 
 
 - K sn ( 
 
 " + K SlU 
 
 for < and <" and when K = 1, 
 
 (9 a) M 2 < K sin 2 < = 2 < sin 2 < (H being here zero). 
 
 These solutions may be made most conveniently by means of tables giving the values of M 
 and H for suitably chosen values of < and <j>', from which approximate values of < and <' may 
 be interpolated, which may then be corrected by the following differential formulae : 
 
 (10) 
 
 S<i = 
 
 2tg. 
 
 sn 
 
 + 
 
 and S < = 
 
 2 K tg $' (sin <' i 
 
 - , f or K < 1. 
 
A STUDY OF THE LIGHT CURVE 
 
 When K > 1, we shall have to use instead of the latter, 
 
 8/7 
 
 (11) 
 
 2 K tg </>' sin ($ <') 
 
 If it be desired to assume a value of q a little greater than unity, it will then be necessary to 
 compute X from equation (5) above. Differentiating (5) with respect to q, I obtain 
 
 d A. _ in (1 + c) 
 
 ~ = ~, 77' 
 
 (I q (m qY 
 
 an essentially negative magnitude. 
 
 X and q therefore change against each other, so that an increase in q will necessitate a decrease 
 in X. Again designating the maximum value of K 2 by K 2 and the minimum by k 2 we have 
 
 m q 
 cq 
 
 and K i = 
 
 m c q 
 
 Differentiating these with respect to q, we find, 
 
 Ct rC VYIi 
 
 dq c q 2 
 
 and 
 
 d q 
 
 = + 
 
 m 
 
 (m 
 
 The former of these differential coefficients is essentially negative, and the latter is essen- 
 tially positive. An augmentation of q will therefore depress the minor and elevate the major 
 limit of K 2 ; and to be able to include a value of q somewhat larger than unity, values of M and 
 If were also computed for K = 0.70. The table of computed M's and H's is given here. 
 
 AUXILIARY TABLES FOR INTERPOLATING APPROXIMATE VALUES OF <f> AND </>". 
 
 <(> for K <1 
 or 
 "for/c>l 
 
 K = 0.70 
 
 K = 0.80 
 
 K = 0.85 
 
 K= 1.15 
 
 K = 1.35 
 
 M 
 
 H 
 
 M 
 
 n 
 
 M 
 
 n 
 
 M 
 
 H 
 
 M 
 
 n 
 
 o / 
 
 
 
 
 
 
 
 
 
 
 
 00 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 2 00 
 
 0.0001 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0001 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 4 00 
 
 0.0005 
 
 0.0001 
 
 0.0006 
 
 0.0001 
 
 0.0005 
 
 0.0003 
 
 0.0007 
 
 0.0001 
 
 0.0009 
 
 0.0001 
 
 6 00 
 
 0.0018 
 
 0.0007 
 
 0.0017 
 
 0.0005 
 
 0.0017 
 
 0.0003 
 
 0.0022 
 
 0.0003 
 
 0.0031 
 
 0.0010 
 
 8 00 
 
 0.0043 
 
 0.0009 
 
 0.0040 
 
 0.0006 
 
 0.0038 
 
 0.0005 
 
 0.0051 
 
 0.0007 
 
 0.0077 
 
 0.0014 
 
 10 00 
 
 0.0086 
 
 0.0013 
 
 0.0080 
 
 0.0009 
 
 0.0077 
 
 0.0008 
 
 0.0100 
 
 0.0008 
 
 0.0152 
 
 0.0021 
 
 12 00 
 
 0.0148 
 
 0.0029 
 
 0.0137 
 
 0.0017 
 
 0.0133 
 
 0.0014 
 
 0.0171 
 
 0.0013 
 
 0.0261 
 
 0.0041 
 
 14 00 
 
 0.0319 
 
 0.0044 
 
 0.0216 
 
 0.0025 
 
 0.0210 
 
 0.0018 
 
 . 0.0274 
 
 0.0019 
 
 0.0414 
 
 O.OOG8 
 
 16 00 
 
 0.0353 
 
 0.0065 
 
 0.0325 
 
 0.0037 
 
 0.0314 
 
 0.0026 
 
 0.0405 
 
 0.0026 
 
 0.0618 
 
 0.0090 
 
 18 00 
 
 0.0503 
 
 0.0093 
 
 0.0461 
 
 0.0054 
 
 0.0444 
 
 0.0038 
 
 0.0577 
 
 0.0038 
 
 0.0880 
 
 0.0148 
 
 20 00 
 
 0.0690 
 
 0.0136 
 
 0.0631 
 
 0.0076 
 
 0.0610 
 
 0.0056 
 
 0.0792 
 
 0.0062 
 
 0.1207 
 
 0.0201 
 
 22 00 
 
 0.0933 
 
 0.0202 
 
 0.0838 
 
 0.0104 
 
 0.0805 
 
 0.0072 
 
 0.1050 
 
 0.0082 
 
 0.1607 
 
 0.0274 
 
 24 00 
 
 0.1195 
 
 0.0247 
 
 0.1087 
 
 0.0142 
 
 0.1043 
 
 0.0096 
 
 0.1306 
 
 0.0107 
 
 0.2086 
 
 0.0300 
 
 26 00 
 
 0.1522 
 
 0.0328 
 
 0.1378 
 
 0.0184 
 
 0.1320 
 
 0.0126 
 
 0.1735 
 
 0.0129 
 
 0.2648 
 
 0.0473 
 
 28 00 
 
 0.1908 
 
 0.0422 
 
 0.1715 
 
 0.0231 
 
 0.1644 
 
 0.0161 
 
 0.2137 
 
 0.0175 
 
 0.3112 
 
 0.0613 
 
 30 00 
 
 0.2344 
 
 0.0542 
 
 0.2104 
 
 0.0293 
 
 0.2014 
 
 0.0280 
 
 0.2020 
 
 0.0228 
 
 0.4079 
 
 0.0779 
 
 32 00 
 
 0.2876 
 
 0.0693 
 
 0.2549 
 
 0.0367 
 
 0.2433 
 
 0.0251 
 
 0.3163 
 
 0.0281 
 
 0.4959 
 
 0.0983 
 
 34 00 
 
 0.3307 
 
 0.0710 
 
 0.3053 
 
 0.0458 
 
 0.2906 
 
 0.0309 
 
 0.4022 
 
 0.0343 
 
 0.5969 
 
 0.1240 
 
 36 00 
 
 0.4176 
 
 0.1120 
 
 0.3621 
 
 0.0555 
 
 0.3436 
 
 0.0381 
 
 0.4457 
 
 0.0425 
 
 0.7116 
 
 0.1548 
 
 38 00 
 
 0.4998 
 
 0.1438 
 
 0.4055 
 
 0.0696 
 
 0.4024 
 
 0.0465 
 
 0.5222 
 
 0.0518 
 
 0.8447 
 
 0.1972 
 
 40 00 
 
 0.5977 
 
 0.1863 
 
 0.4968 
 
 0.0853 
 
 0.4679 
 
 0.0569 
 
 0.6059 
 
 0.0022 
 
 0.9942 
 
 0.2446 
 
OF THE VARIABLE STAR U PEGASI. 
 
 <(> for K <1 
 or 
 <f," for K >1 
 
 K 0.70 
 
 K = 0.80 
 
 K = 0.85 
 
 K = l.K 
 
 K 1.36 
 
 M 
 
 H 
 
 M 
 
 H 
 
 M 
 
 H 
 
 M 
 
 H 
 
 M 
 
 B 
 
 o / 
 
 
 
 
 
 
 
 
 
 
 
 42 00 
 
 0.7218 
 
 0.2503 
 
 0.5772 
 
 0.1047 
 
 0.5397 
 
 0.0681 
 
 0.6984 
 
 0.0752 
 
 1.1765 
 
 0.3103 
 
 43 00 
 
 0.8028 
 
 0.2994 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.2694 
 
 0.3591 
 
 43 30 
 
 0.8460 
 
 0.3328 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 44 00 
 
 0.9281 
 
 0.3818 
 
 0.6658 
 
 0.1296 
 
 0.6162 
 
 0.0858 
 
 0.7996 
 
 0.0908 
 
 1.3790 
 
 0.4010 
 
 44 20 
 
 0.9797 
 
 0.4388 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 44 26 
 
 1.0297 
 
 0.4783 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 45 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.5036 
 
 0.4638 
 
 46 00 
 
 
 
 
 
 0.7666 
 
 0.1600 
 
 0.7066 
 
 0.1002 
 
 0.9103 
 
 0.1087 
 
 1.6499 
 
 0.5445 
 
 47 00 
 
 
 
 
 
 0.8229 
 
 0.1731 
 
 
 
 
 
 
 
 
 
 1.8417 
 
 0.6700 
 
 47 30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.9983 
 
 0.7763 
 
 47 48 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2.1843 
 
 0.9573 
 
 48 00 
 
 
 
 
 
 0.8810 
 
 0.1999 
 
 0.8028 
 
 0.1217 
 
 1.0308 
 
 0.1303 
 
 
 
 
 
 49 00 
 
 
 
 
 
 0.9488 
 
 0.2288 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50 00 
 
 
 
 
 
 1.0215 
 
 0.2009 
 
 0.9094 
 
 0.1490 
 
 1.1641 
 
 0.1578 
 
 _ 
 
 
 
 51 00 
 
 
 
 __ 
 
 1.1055 
 
 0.3032 
 
 
 
 
 
 
 
 
 
 
 
 
 
 51 30 
 
 
 
 
 
 1.1534 
 
 0.3301 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 52 00 
 
 
 
 _ 
 
 1.2082 
 
 0.3632 
 
 1.0290 
 
 0.1842 
 
 1.3129 
 
 0.1929 
 
 
 
 
 
 52 30 
 
 
 
 
 
 1.2726 
 
 0.4083 
 
 
 
 
 
 
 
 
 
 
 
 
 
 53 00 
 
 
 
 
 
 1.3736 
 
 0.4847 
 
 
 
 
 
 
 
 
 
 
 
 
 
 53 8.7 
 
 
 
 
 
 1.4531 
 
 0.5575 
 
 
 
 
 
 
 
 
 
 
 
 
 
 54 00 
 
 
 
 
 
 
 
 
 
 1.1675 
 
 0.2480 
 
 1.4749 
 
 0.2376 
 
 
 
 
 
 56 00 
 
 
 
 
 
 
 
 
 
 1.3320 
 
 0.3044 
 
 1.6574 
 
 0.2991 
 
 
 
 
 
 57 00 
 
 
 
 
 
 
 
 
 
 1.4389 
 
 0.3728 
 
 
 
 
 
 
 
 
 
 57 30 
 
 
 
 
 
 
 
 
 
 1.5066 
 
 0.4060 
 
 
 
 
 
 
 
 
 
 58 00 
 
 
 
 
 
 
 
 
 
 1.5992 
 
 0.4734 
 
 1.8764 
 
 0.3874 
 
 
 
 
 
 58 12.7 
 
 
 
 
 
 
 
 
 
 1.7040 
 
 0.5666 
 
 
 
 
 
 
 
 
 
 58 30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.9414 
 
 0.4192 
 
 _ 
 
 
 
 59 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2.0187 
 
 0.4567 
 
 
 
 
 
 59 30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2.0970 
 
 0.5050 
 
 __ 
 
 
 
 CO 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2.2032 
 
 0.5786 
 
 
 
 
 
 60 25 
 
 ~~ 
 
 
 
 
 
 
 
 
 
 
 
 2.3976 
 
 0.7439 
 
 
 
 
 
 The table gives the values of M and ff, of course, only up to </> sin *, or to <j>' = sin -> 
 according as ^ 1. 
 
 The expression for AT when K = 1 is so simple as to render the use of an auxiliary table un- 
 necessary, and this case has therefore not been included in the foregoing lists. 
 M and H are connected with the observations by means of the relations : 
 
 (12) M = TT (1 + K 2 X) (1 - /) and H = IT K a - * (1 + * 2 A.) (1 - J), 
 
 where J is obtained from the light curve by subtracting the ordinate of the curve for any 
 given instant from the mean ordinate for the maxima, calling this difference A(r and substituting 
 in the equation : 
 
 log J = 0.04 A G (\G being in tenths of a magnitude). 
 
 The values of M and H, on the various hypotheses for K and for the times preceding and 
 following Min. I. given in the first column, are tabulated here. 
 
A STUDY OF THE LIGHT CURVE 
 
 VALUES OF M AND H COMPUTED FROM THE LIGHT CURVE FOR THE EPOCHS t. 
 
 t 
 
 K = 0.80 
 
 K = 0.85 
 
 K = 1.00 
 
 K=115 
 
 K = 1.35 
 
 Jt 
 
 H 
 
 M 
 
 H 
 
 M 
 
 H 
 
 M 
 
 H 
 
 M 
 
 H 
 
 h. 
 
 
 
 
 
 
 
 
 
 
 
 -1.50 
 
 0.1624 
 
 1.8482 
 
 0.1694 
 
 2.1004 
 
 0.1930 
 
 2.9486 
 
 0.2203 
 
 3.9345 
 
 0.2628 
 
 5.4628 
 
 -1.25 
 
 0.3910 
 
 1.6196 
 
 0.4077 
 
 1.8621 
 
 0.4625 
 
 2.6771 
 
 0.5303 
 
 3.6245 
 
 0.6325 
 
 5.0931 
 
 -1.00 
 
 0.6903 
 
 1.3203 
 
 0.7200 
 
 1.5498 
 
 0.8201 
 
 2.3215 
 
 0.9364 
 
 3.2184 
 
 1.1168 
 
 4.G088 
 
 -0.75 
 
 1.0671 
 
 0.9435 
 
 1.1129 
 
 0.1569 
 
 1.2677 
 
 1.8739 
 
 1.4476 
 
 2.7072 
 
 1.7264 
 
 3.9992 
 
 -0.50 
 
 1.4496 
 
 0.5610 
 
 1.5118 
 
 0.7580 
 
 1.7221 
 
 1.4195 
 
 1.9664 
 
 2.1884 
 
 2.3452 
 
 3.3804 
 
 -0.25 
 
 1.8458 
 
 0.1648 
 
 1.9250 
 
 0.3448 
 
 2.1927 
 
 0.9489 
 
 2.5038 
 
 2.6510 
 
 2.9861 
 
 2.7395 
 
 -0.12| 
 
 1.9067 
 
 0.1039 
 
 1.9885 
 
 0.2813 
 
 2.2650 
 
 0.8766 
 
 2.5864 
 
 1.5684 
 
 3.0847 
 
 2.6409 
 
 0.00 
 
 1.9326 
 
 0.0780 
 
 2.0156 
 
 0.2542 
 
 2.2959 
 
 0.8457 
 
 2.6216 
 
 1.5332 
 
 3.1267 
 
 2.5989 
 
 + 0.12 
 
 1.8986 
 
 0.1120 
 
 1.9801 
 
 0.2897 
 
 2.2555 
 
 0.8861 
 
 2.5755 
 
 1.5793 
 
 3.0727 
 
 2.6529 
 
 + 0.25 
 
 1.7192 
 
 0.2914 
 
 1.7930 
 
 0.4768 
 
 2.0423 
 
 1.0993 
 
 2.3321 
 
 1.8227 
 
 2.7814 
 
 2.9442 
 
 + 0.50 
 
 1.2380 
 
 0.7726 
 
 1.2912 
 
 0.9786 
 
 1.4708 
 
 1.6708 
 
 1.6794 
 
 2.4754 
 
 2.0030 
 
 3.7226 
 
 + 0.75 
 
 0.8358 
 
 1.1758 
 
 0.8716 
 
 1.3982 
 
 0.9928 
 
 2.1488 
 
 1.1337 
 
 3.0211 
 
 1.3521 
 
 4.3735 
 
 + 1.00 
 
 0.5383 
 
 1.4623 
 
 0.5614 
 
 1.7084 
 
 0.6395 
 
 2.5021 
 
 0.7302 
 
 3.4246 
 
 0.8708 
 
 4.8548 
 
 + 1.25 
 
 0.2748 
 
 1.7258 
 
 0.2866 
 
 1.9832 
 
 0.3265 
 
 2.8151 
 
 0.3728 
 
 3.7820 
 
 0.4446 
 
 5.2810 
 
 + 1.50 
 
 -0.0642 
 
 1.9464 
 
 0.0670 
 
 2.2028 
 
 0.0763 
 
 3.0653 
 
 0.0871 
 
 4.0677 
 
 0.1039 
 
 5.6217 
 
 (13) 
 
 The distance of centres, r, is seen from the accompanying figure to be given by 
 
 * 8ln *" 
 
 p = 
 
 sin 
 
 where sin = K sin #> and 
 
 = 180. 
 
 The figure relates only to the case in which K <1 and <f>" <90, but the modifications 
 necessary to adapt it to the cases where K >1 and </>" <90, are so obvious, that they may 
 be left to the reader. 
 
 Assuming now a circular orbit, and denoting by a and ft + -, the longitude in the 
 
 2t 
 
 apparent orbit and the true anomaly in the real orbit respectively, both counted from the 
 node, and calling r and p the radii vectorcs in the true and apparent orbits, we may write, 
 
 p cos a = r sin ft and tg a = cos i cot /?, 
 p a = r 2 sin 2 J3 + r 2 cos 2 i cos 2 /?. 
 
 whence, 
 
 (14) 
 
 Calling, for brevity, 
 
 (14 a) x = r 2 and y = r 2 cos 2 i = x cos 2 i, 
 
 and we then have the following simple relation between the various magnitudes ; 
 
 (15) p 2 = x sin* ft + y cos 2 ft. (ft = p. t 40 t, t in hours from Min. I.), 
 
 which holds for all cases except when the smaller disc is projected wholly upon the larger 
 at the epoch of Min. I. 
 
 The solution of equations (8), (9) and (9a) for the five hypothetical values of K gave 
 the results here tabulated. 
 
OF THE VARIABLE STAR U PEGASI. 
 
 9 
 
 TABULATED VALUES OF <f> AND p. 
 
 t 
 
 K =: 
 
 D.80 
 
 K 
 
 D.85 
 
 K 
 
 LOO 
 
 K = 
 
 1.15 
 
 K =z 
 
 1.36 
 
 
 <*> 
 
 P 
 
 * 
 
 P 
 
 * 
 
 P 
 
 W 
 
 P 
 
 $H 
 
 P 
 
 
 o / 
 
 
 / 
 
 
 / 
 
 
 / 
 
 
 / 
 
 
 -1.50 
 
 27 30.3 
 
 1.5403 
 
 28 18.5 
 
 1.5440 
 
 30 40.0 
 
 1.7202 
 
 28 17.0 
 
 1.8510 
 
 25 56.0 
 
 2.0210 
 
 -1.25 
 
 36 56.0 
 
 1.3276 
 
 38 10.4 
 
 1.3697 
 
 41 46.5 
 
 1.4916 
 
 38 12.5 
 
 1.6066 
 
 34 39.0 
 
 1.7515 
 
 -1.00 
 
 44 30.5 
 
 1.0987 
 
 46 17.5 
 
 1.1381 
 
 51 25.5 
 
 1.2472 
 
 46 26.6 
 
 1.3449 
 
 41 15.0 
 
 1.4710 
 
 -0.75 
 
 50 33.0 
 
 0.8440 
 
 53 16.0 
 
 0.8813 
 
 60 45.0 
 
 0.9774 
 
 53 43.0 
 
 1.0557 
 
 46 26.2 
 
 1.1379 
 
 -0.50 
 
 53 8.0 
 
 0.6012 
 
 57 32.1 
 
 0.6397 
 
 68 43.0 
 
 0.7260 
 
 58 40.3 
 
 0.7848 
 
 36 18.0 
 
 0.5558 
 
 -0.25 
 
 46 12.6 
 
 0.3482 
 
 56 44.0 
 
 0.3951 
 
 76 8.5 
 
 0.4790 
 
 45 51.0 
 
 0.2360 
 
 47 35.0 
 
 0.8292 
 
 -0.12 
 
 41 52.0 
 
 0.3038 
 
 55 28.6 
 
 6.3577 
 
 77 14.0 
 
 0.4420 
 
 51 47.3 
 
 0.2833 
 
 47 46.2 
 
 0.8805 
 
 0.00 
 
 39 8.0 
 
 0.2841 
 
 54 44.0 
 
 0.3411 
 
 77 42.0 
 
 0.4260 
 
 53 29.0 
 
 0.3024 
 
 47 47.2 
 
 0.8897 
 
 + 0.12 
 
 42 40.0 
 
 0.3101 
 
 55 40.0 
 
 0.3626 
 
 77 5.5 
 
 0.4474 
 
 51 10.3 
 
 0.2771 
 
 47 45.4 
 
 0.8726 
 
 + 0.25 
 
 50 45.2 
 
 0.4321 
 
 58 1.4 
 
 0.4745 
 
 73 50.0 
 
 0.5568 
 
 60 21.8 
 
 0.6006 
 
 46 28.8 
 
 0.7254 
 
 + 0.50 
 
 52 13.0 
 
 0.7342 
 
 55 33.3 
 
 0.7718 
 
 64 26.3 
 
 0.8630 
 
 56 14.0 
 
 0.9326 
 
 47 33.3 
 
 0.9982 
 
 + 0.75 
 
 47 13.6 
 
 0.9969 
 
 49 19.0 
 
 1.0356 
 
 55 16.1 
 
 1.1394 
 
 49 33.3 
 
 1.2298 
 
 43 45.5 
 
 1.3331 
 
 + 1.00 
 
 41 4.3 
 
 1.1781 
 
 42 34.0 
 
 1.2512 
 
 46 54.4 
 
 1.3664 
 
 42 39.0 
 
 1.4729 
 
 38 23.0 
 
 1.6037 
 
 + 1.25 
 
 32 49.0 
 
 1.4288 
 
 30 50.2 
 
 1.4729 
 
 36 51.0 
 
 1.6004 
 
 33 50.0 
 
 1.7226 
 
 30 52.0 
 
 1.8803 
 
 + 1.50 
 
 
 
 20 40.0 
 
 1.7093 
 
 22 18.1 
 
 1.8518 
 
 20 40.0 
 
 1.9909 
 
 19 1.0 
 
 2.2309 
 
 
 
 
 
 
 
 
 
 
 
 
 A comparison of the values of p on the last two hypotheses for K, shows at once that 
 these values of K need not be further considered, since the values of p in both cases fall 
 for a time, reach a minimum before Min. I., rise to a maximum value about the time t = 0, 
 fall to a second minimum value, and then rise continuously ; and since p denotes the radius 
 vector of the apparent orbit, which latter must be an ellipse, obviously such a variation of 
 it must be impossible. The value of K, i.e., the radius of the darker body cannot, therefore, 
 have either of these latter values. 
 
 Substituting the values of p for the first three assumptions for K in equation (15) above, 
 we shall have the following 15 observation equations : 
 
 OBSERVATION EQUATIONS. 
 
 
 (e = 0.80 
 
 K = 085 
 
 K=1.00 
 
 r yx for 
 K = 0.80, K = 0.85, K 1.00 
 
 ( 1) 0.7500 a; + 0.2500 y 
 
 =2.3725 ; 
 
 =2.3839; 
 
 =2.9588 
 
 1.4915 
 
 1.7764 
 
 2.0207 
 
 ( 2) 0.5868 +0.4132 
 
 =1.7625 ; 
 
 =1.8761 ; 
 
 =2.2249 
 
 1.7561 
 
 1.8052 
 
 1.9543 
 
 ( 3) 0.4132 -t 0.5868 
 
 =1.2071 ; 
 
 =1.2953; 
 
 =1.5555 
 
 1.7281 
 
 1.7776 
 
 1.9270 
 
 ( 4) 0.2500 +0.7500 
 
 =0.7123 ; 
 
 =0.7767 ; 
 
 =0.9553 
 
 1.6411 
 
 1.6878 
 
 1.8358 
 
 ( 5) 0.1170 +0.8830 
 
 =0.3614; 
 
 =0.4092 ; 
 
 =0.5271 
 
 1.6746 
 
 1.7274 
 
 1.8767 
 
 ( 6) 0.0302 +0.9698 
 
 =0.1213; 
 
 =0.1561; 
 
 =0.2294 
 
 1.1693 
 
 1.1693 
 
 1.2300 
 
 ( 7) 0.0076 +0.9924 
 
 =0.0923 ; 
 
 =0.1279; 
 
 =0.1954 
 
 1.2670 
 
 1.2825 
 
 1.4142 
 
 ( 8) 0.0000 +1.0000 
 
 =0.0807 ; 
 
 =0.1163; 
 
 =0.1815 
 
 1.4545 
 
 1.4554 
 
 1.6264 
 
 ( 9) 0.0076 +0.9924 
 
 =0.0962 ; 
 
 =0.1315; 
 
 =0.2002 
 
 2.0176 
 
 2.0610 
 
 2.2410 
 
 (10) 0.0302 +0.9698 
 
 =0.1867; 
 
 =0.2252 ; 
 
 =0.3100 
 
 2.0303 
 
 1.0893 
 
 2.2417 
 
 (11) 0.1170 +0.8830 
 
 =0.5390 ; 
 
 =0.5957 ; 
 
 =0.7448 
 
 1.8277 
 
 1.9393 
 
 2.1018 
 
 (12) 0.2500 +0.7500 
 
 =0.9938 ; 
 
 =1.0727 ; 
 
 =1.2982 
 
 1.6723 
 
 1.8275 
 
 1.9780 
 
 (13) 0.4132 +0.5868 
 
 =1.3879; 
 
 =1.5655; 
 
 =1.8670 
 
 1.8966 
 
 1.8992 
 
 2.0524 
 
 (14) 0.5868 +0.4132 
 
 =2.0415; 
 
 =2.1694; 
 
 =2.5613 
 
 
 
 2.0184 
 
 2.1772 
 
 (15) 0.7500 +0.2500 
 
 ^^ ~ 
 
 =2.9217; 
 
 =3.4292 
 
 
 
 
10 A STUDY OF THE LIGHT CURVE 
 
 Each pair of these equations furnishes a value for both x and /, and from the rcsulls 
 of their solution the values of r and cos 2 i may be obtained with the help of (14a). The 
 assumption of a circular form of the orbit requires that the different values of r and of 
 cos 2 z, on the correct hypothesis for K, shall all be approximately equal. The values for r 
 obtained by solving (1) and (2), (2) and (3), (3) and (4), etc., in succession for the various 
 values of K are tabulated in the last three columns of the foregoing table. The mean 
 values and probable errors for each of the assumptions for K are : for K = 0.80, r = 1.6636 
 0.0485 ; for K = 0.85, r = 1.7512 0.0494, and for * = 1.00, r = 1.9341 0.0535. The indi- 
 vidual determinations of cos 2 i are not given here, but the corresponding means and probable 
 errors are, for the respective cases : 
 
 cos 2 * = +0.0275 0.0069; = +0.0482 0.0072; = +0.0547 0.0074. 
 
 The difference of the probable errors is not great in any case, but both r and cos 2 i agree in 
 their testimony favoring the smallest value of K as being the most probable. Assuming this value 
 of K however, a physical peculiarity, though not an impossibility, is met in the circumstance that 
 the most probable distance of centres (1.6634) is considerably less than the sum of the radii 
 (=1.8), i. e., the masses must interpenetrate, and consequently form a single body (Poincard's 
 apiod). 
 
 The probable errors not differing by enough to enable them to pronounce with sufficient 
 emphasis for any one of the hypotheses, it seemed desirable to approach the problem also in- 
 directly to see whether the conclusions will be the same as those given by this direct solution. 
 That the foregoing discussion, however, indicates conclusively that the correct value of K is 
 smaller than 0.85, there can be no doubt. 
 
 INDIRECT SOLUTION. 
 
 The mode of procedure here is to read from the light curve for suitably chosen epochs, 
 the instantaneous brightnesses in stellar magnitudes, to form the differences between these 
 brightnesses and the maximum brightness, to convert these differences, by means of the 
 Pogson scale, into their equivalent light ratios, to compare these ratios with the corresponding 
 ratios, computed from certain assumed elements, and finally, after finding sufficiently close 
 approximations to the correct values of the elements, to adjust these differences in the sense 
 computation minus observation, by the method of Least Squares. 
 
 Letting J' and J" denote the instantaneous brightnesses in the neighborhood of Min. I. and 
 Min. II. respectively, and M 1 , H', M", and If", the corresponding values of the M and // defined 
 by equations (12), it will be seen by referring to my article on Beta Lyrae, in the January 
 Astrophysical Journal, that 
 
 (10 -** 
 
 and hence, there is an obvious advantage in adjusting 1 J 1 and 1 J"" instead of J' and J". 
 The former quantities were therefore used throughout the reductions. 
 The equations for computing M', or M 1 ' are : 
 ( (a) jS = 40 t. 
 
 (b) p = r \/sm 2 ft + cos 2 i cos 2 /3. If = i' is near - , t' is small and 
 
 
 
 (c)o r -v/sin 2 8 + i n cos 2 B. If i' = o, p = r sin /3. 
 
 (17) 
 
 (d) cos <t> = 
 
 1 + p 2 - >c 2 
 
 2 P 
 
 (e) sin </>' = - sin <. 
 (/) M 1 , or M" - < + /< 2 <j>" - K sin (< + <") = < + 2 </>" - p sin 
 
OF THE VARIABLE STAR U PEGASI. 
 
 11 
 
 These, together with (16), determine 1 /' and 1 J" from the light curve. The value 
 of cos 2 i as found above, was small, and as a first approximation i was taken -, or i' = 90 t = 0. 
 
 To neglect the effect of orbital eccentricity requires Min. II. to fall at the middle point 
 of the period. Disregarding provisionally the slight displacement of this chief epoch from 
 the middle point, taking ordinates equidistant from Min. I. and Min. II. before and after 
 these epochs, forming the means for each epoch separately and computing the corresponding 
 values of 1 J' and 1 J", the results here tabulated were obtained. 
 
 t 
 
 Minimum T. 
 
 Minimum II. 
 
 i-j'. 
 
 \-j". 
 
 Before. 
 
 After. 
 
 Mean. 
 
 Before. 
 
 After. 
 
 Mean. 
 
 1.50 
 
 9.32 
 
 9.34 
 
 9.33 
 
 9.36 
 
 9.35 
 
 9.35 
 
 0.0228 
 
 0.0000 
 
 1.25 
 
 9.37 
 
 9.37 
 
 9.37 
 
 9.41 
 
 9.40 
 
 9.40 
 
 0.0742 
 
 0.0362 
 
 1.00 
 
 9.42 
 
 9.44 
 
 9.43 
 
 9.50 
 
 9.47 
 
 9.48 
 
 0.1330 
 
 0.0896 
 
 0.75 
 
 9.51 
 
 9.53 
 
 9.52 
 
 9.61 
 
 9.55 
 
 9.58 
 
 0.2072 
 
 0.1606 
 
 0.50 
 
 9.62 
 
 9.64 
 
 9.63 
 
 9.73 
 
 9.67 
 
 9.70 
 
 0.2901 
 
 0.2380 
 
 0.25 
 
 9.72 
 
 9.72 
 
 9.72 
 
 9.84 
 
 9.84 
 
 9.84 
 
 0.3749 
 
 0.3018 
 
 0.00 
 
 9.75 
 
 9.75 
 
 9.75 
 
 9.90 
 
 9.90 
 
 9.90 
 
 0.4084 
 
 0.3208 
 
 The results of this table are shown graphically on Plate II. 
 
 The values of M' and M" for various assumptions for K both greater and less than 
 the minimum value given above (0.7755) were computed from formula (16). For values 
 of K less than 0.7755, it was of course necessary to assume a flattening of the discs and 
 
 nyi _ ft ft 
 
 to compute X from the formula X = - , (<? depending on the flattening). After having 
 
 computed a number of light curves for various assumptions for r and K, it became evident 
 that the discs must lie extremely close together. The attempt was then made to ascertain 
 what value of K would best satisfy the observations on the hypothesis of contact of discs. 
 The residuals M M c = A If were formed for the values of t in the above table and 
 compared, that hypothesis furnishing the smallest mean residual, A M, being assumed to 
 lie nearest the truth. M was computed from the observed values of J 1 and J" by the aid 
 of the formulas, 
 
 M' = (1 + K 2 X) TT (1 - J') ; M" = (1 + K 2 A)(1- J"); M = 1/2 (M' + M") 
 
 A 
 
 and M c by the aid of 
 
 M e = <f> + K 2 <" p' sin <, p' = - p and - = A/sin 8 /3 + </ cos 2 ft. 
 
 J J 
 
 The value K = 0.8, involving a somewhat simpler hypothesis than K = 0.75, viz : q = 1, 
 was taken as a basis for further experimentation, notwithstanding the fact that the latter 
 value of K gave a somewhat smaller mean residual. If, moreover, as seems now probable, 
 a part of the light change be ascribed to a flattening, the larger value of K will probably be 
 nearer the truth. The distance of centres, r, was then assumed to be 1 + 1.1 K and 1 + 0.9 K 
 in turn, and the mean residuals computed on these hypotheses. The results of these six 
 hypotheses are here tabulated. Unless otherwise explicitly stated, it is to be understood 
 that r = 1-f K, that q = 1, and that the components are similar ellipsoids of revolution. 
 For the case where K 0.75, the smallest possible value of q (= 1.0271) was used in 
 obtaining M . 
 
12 
 
 A STUDY OF THE LIGHT CURVE 
 
 VALUES OF J/, 31,. AND A M ON VARIOUS HYPOTHESES FOR 
 
 Jt/o Me A M 
 
 SI M a A M 
 
 K = 0.75, q = 1.0271 
 
 K = 0.8 
 
 0.0514 
 
 0.1372 
 
 -0.0858 
 
 0.0538 
 
 0.1456 
 
 -0.0918 
 
 0.2728 
 
 0.2926 
 
 -0.0248 
 
 0.2839 
 
 0.3303 
 
 -0.0464 
 
 0.5612 
 
 0.5720 
 
 -0.0108 
 
 0.5732 
 
 0.6097 
 
 -0.0265 
 
 0.9358 
 
 0.9189 
 
 + 0.0169 
 
 0.9746 
 
 0.9810 
 
 -0.0064 
 
 1.3486 
 
 1.3300 
 
 + 0.0186 
 
 1.3998 
 
 1.4274 
 
 -0.0276 
 
 1.7259 
 
 1.7299 
 
 -0.0040 
 
 1.7915 
 
 1.8949 
 
 -0.1034 
 
 Mean residual 0.0302 
 
 Mean residual 0.0503 
 
 K = 0.9 
 
 K = 0.95 
 
 0.0586 
 
 0.1634 
 
 -0.1048 
 
 0.0612 
 
 0.1690 
 
 -0.1078 
 
 0.3092 
 
 0.3703 
 
 -0.0611 
 
 0.3229 
 
 0.3911 
 
 -0.0682 
 
 0.6349 
 
 0.6850 
 
 -0.0501 
 
 0.0632 
 
 0.7228 
 
 -0.0596 
 
 1.0579 
 
 1.1051 
 
 -0.0472 
 
 1.1049 
 
 1.0957 
 
 -0.0092 
 
 1.5242 
 
 1.6394 
 
 -0.1152 
 
 1.5919 
 
 1.7098 
 
 -0.1179 
 
 1.9509 
 
 2.1904 
 
 -0.2395 
 
 2.0375 
 
 2.3206 
 
 -0.2831 
 
 Mean residual 0.1030 
 
 Mean residual 0.1076 
 
 K = 0.8, r = 1 + 0.9 it 
 
 = 0.8, r 1 + 1.1 K 
 
 0.0538 
 
 0.2116 
 
 -0.1578 
 
 0.0538 
 
 0.0882 
 
 -0.0344 
 
 0.2839 
 
 0.4029 
 
 -0.1190 
 
 0.2839 
 
 0.2672 
 
 + 0.0167 
 
 0.5732 
 
 0.6806 
 
 -0.1074 
 
 0.5732 
 
 0.5179 
 
 + 0.0553 
 
 0.9746 
 
 1.0313 
 
 -0.0567 
 
 0.9746 
 
 0.8897 
 
 + 0.0849 
 
 1.3998 
 
 1.6714 
 
 -0.2716 
 
 1.3998 
 
 1.3835 
 
 + 0.0163 
 
 1.7915 
 
 1.9134 
 
 -0.1219 
 
 1.7915 
 
 1.8760 
 
 -0.0845 
 
 Mean residual 0.1391 
 
 Mean residual 0.0487 
 
 The low value of the mean residual of the first hypothesis, viz. : K = 0.75 and q = 1.0271, 
 renders a further investigation into the general effect of a flattening desirable. The light 
 curves for y = 1.01, 1.02, 1.03, 1.04 and 1.05 were now computed and the residuals, AJ!/^, 
 formed with the proper values of X, from X= (TO c q) / (m q), with K = 0.7785 and r = 1.7814. 
 
 VALUES OF AM _ C FOR VARIOUS VALUES OF q. 
 
 5=1.01 
 
 q = 1.02 
 
 q = 1.03 
 
 9 = 1.04 
 
 ? = 1.06 
 
 -0.0835 
 
 -0.0607 
 
 -0.0774 
 
 -0.0737 
 
 -0.0705 
 
 -0.0182 
 
 -0.0274 
 
 -0.0086 
 
 -0.0017 
 
 + 0.0046 
 
 -0.0031 
 
 + 0.0018 
 
 + 0.0295 
 
 + 0.0377 
 
 + 0.0483 
 
 + 0.0229 
 
 + 0.0244 
 
 + 0.0557 
 
 + 0.0676 
 
 + 0.0788 
 
 + 0.0149 
 
 + 0.0113 
 
 + 0.0389 
 
 + 0.0493 
 
 + 0.0609 
 
 -0.0409 
 
 -0.0533 
 
 -0.0391 
 
 -0.0330 
 
 -0.0262 
 
 Si. 0.0306 
 
 0.0298 
 
 0.0415 
 
 0.0438 
 
 0.0482 
 
OF THE VARIABLE STAB U PEGASI. 13 
 
 These latter values of K and r resulted from a Least Square solution of a set of observation 
 equations connecting d /c, d r, and d i' (_ d i) , which gave d i' = V 0.0096. This value of d i' 
 being imaginary but small, it was put = 0. The residuals for the five hypotheses are col- 
 lected in the table last preceding. 
 
 For the case in which q = 1.00, the third column for K = 0.8 above may be examined. 
 
 Both the run of the individual values of A M _ e and the magnitude of the mean residual 
 indicate q = 1.02 to be most approximate. 
 
 Differential equations were now derived in such form as to connect dk, dr, dq, and di' z , 
 with dM^.M _,.. The derivation of these relations was made as follows: 
 
 Differentiating M = <j> + K 2 </>" - P ' sin <, where P' = (I//) p and (I//) = Vsin 2 ft + q 2 cos 2 /?, 
 
 I rcdu 
 we find, 
 
 n' 2 + K 2 1 1 + p' 2 K 2 p' 2 1 
 
 and reducing by means of cos <f>" =- , cos < = and cos (< + </>") = 
 
 ' 
 
 dM= (1 - p' cos <t>) d<t> + K 2 d<j>" + 2 * <" d K - sin <f> . dp', 
 
 K cos <j>" , . /cosrf>" sin d>"\ 7 cos d> 
 
 dp' and dd>" = [ - --- ) dK --- dp' 
 p' sin </> \p' tan < p 1 ) p 1 sin <f> 
 
 p sin 
 
 which give, after some simplifications, 
 
 dM= 2 <f>" K d - 2 sin <f>dp'. 
 
 But dp 1 = j. dp+p d ^j, where p = r \Xsin 2 /3 + i" cos 2 ^ for small values of 9 = -x i. 
 
 Differentiating and substituting we obtain finally, 
 
 (A) dM=2 K <l> dK-fidr- Qdq-Sdi 1 * 
 in which 
 
 p sin <j> r 
 
 Jl = 2 ^r~ ; Q = 1q pf cos 2 ft sin < ; and I = - sin </> cos 2 /3 esc ft. 
 
 Computing the coefficients 2 K <", R, Q, and Jwith the values q = 1.02, X = 0.7748, = 0.7785, 
 r = 1.7816, and i' o, for the epochs used in the foregoing tables, the following six observation 
 equations were obtained : 
 
 0.9154 dk 
 
 - 0.7517 dr 
 
 - 0.3403 dq 
 
 - 0.2232 di 13 
 
 -0.0607 = 
 
 1.2387 
 
 -0.8591 
 
 - 0.6383 
 
 -0.5389 
 
 -0.0274 = 
 
 1.5827 
 
 -0.8616 
 
 -0.9019 
 
 -1.0897 
 
 + 0.0018 = 
 
 1.9702 
 
 -0.7537 
 
 -1.0031 
 
 -2.0139 
 
 + 0.0244 = 
 
 2.4611 
 
 -0.5420 
 
 -0.8119 
 
 -3.6442 
 
 + 0.0113 = 
 
 3.4477 
 
 -0.2205 
 
 -0.3762 
 
 -6.3170 
 
 -0.0533 = 
 
 These were rendered homogeneous by putting 
 
 dk = 0.2901*; dr = 1.1606 y; dq = 0.9969s; di' a = 0.1583 w, and v = 0.0607. 
 The equations then were : 
 
 0.2655 as 
 
 - 0.8724 y 
 
 -0.33923 
 
 - 0.0353 w 
 
 - 1.0000 v = 
 
 0.3593 
 
 -0.9971 
 
 -0.6363 
 
 -0.0855 
 
 -0.4514 = 
 
 0.4591 
 
 -1.0000 
 
 -0.8991 
 
 -0.1725 
 
 + 0.0297 = 
 
 0.5715 
 
 -0.8747 
 
 -1.0000 
 
 -0.3188 
 
 + 0.4020 = 
 
 0.7138 
 
 -0.6290 
 
 -0.8094 
 
 -0.5769 
 
 + 0.1862 = 
 
 1.0000 
 
 -0.2529 
 
 -0.3750 
 
 -1.0000 
 
 -0.8781 = 
 
14 A STUDY OF THE LIGHT CURVE 
 
 These resulted in the following Normal Equations: 
 
 + 2.2467* - 2.2508 y -2.2557s -1.7l33o - 0.9296 i/=0 
 
 -2.2508 +3.9800 +3.3081 +1.1833 +1.04(32 = 
 
 -2.2557 +3.3081 +3.1241 +1.3822 +0.3763 = 
 
 -1.7133 +1.1833 +1.3822 +1.4727 +0.7113 =0 
 
 The solution of these normals gave 
 
 x = -0.302; y = +0.056; z = -0.183; and w = -0.194 
 and hence, 
 
 dx = -0.087; dr = +0.064; dq = -0.183; and di' 2 = i n = -0.031. 
 
 Inasmuch as an imaginary value of i 1 (= d i') can have no physical significance, the d i' 2 can 
 only be put equal to zero, and the equations solved on this hypothesis gave, 
 
 d = -0.088; dr = -0.125; and dq = -0.170. 
 
 The assumed value of q was 1.02, and consequently the maximum allowable negative value 
 for d q = 0.02, a magnitude considerably smaller than that resulting from the Least Square 
 solution. 
 
 Another important contravention of the physical conditions involved in the problem is that 
 die and dq should be of unlike signs. This can be readily shown. We have seen above that 
 
 K a > m ~ g . Taking the least value of * 2 consistent with physical conditions, viz., 2 = , 
 
 cq dJ m C 3 
 
 and differentiating it, we obtain - = . an essentially negative magnitude. The latter 
 
 d q 2ci<q 2 
 
 relationship would be emphasized more strongly by using K Z greater than this least value. 
 
 While, therefore, a Least Square adjustment can add nothing to the accuracy of the 
 values obtained experimentally, the magnitudes and signs of the values of d K and d r furnished 
 by the adjustment indicate that K and r should be corrected toward the values of these 
 quantities derived in the direct solution of the first part of this paper. Inasmuch as the 
 foregoing results are the best that could be obtained after having computed twenty-five or 
 thirty different light curves in which the elements were shifted in almost every conceivable 
 way, it may be asserted with confidence that the following results may be regarded as the 
 best attainable in the present state of the observational material : 
 
 1. The light curve of U Pegasi given in Harvard College Observatory Circular, No. 23, 
 is satisfactorily represented by the satellite theory. 
 
 2. The distance of centres does not materially differ from the sum of the radii of the 
 components, suggesting the probable concrete existence of the " apiodal " form of Poincare". 
 
 S. The smaller companion is about 0.77 as bright as the larger, and the ratio of radii is 
 approximately 1:0.78. 
 
 4. The inclination of the orbit is very nearly 90, and the disc of one or both bodies, if 
 separate, is slightly flattened. 
 
 5. The accuracy of present observations does not suffice to determine the elements of 
 the " system " completely, since the foregoing discussion shows the residuals to be incapable 
 of adjustment by Least Squares. 
 
 6. The manner of rise and fall of the observed curve after and before the minima, which 
 portions of the curve were determined with especial care, fails to confirm one's first impres- 
 sion on examining the curve, viz. : that the components are separated enough to remain apart 
 for an appreciable time at the maxima. The difference between the durations of uniform 
 brightness at the maxima, as shown by the curve, would seem to indicate a considerable orbital 
 eccentricity, whereas the small distance of centres nullifies the possibility of its existence. It, 
 therefore, seems desirable to direct attention to the importance of a careful photometric study 
 
OF THE VARIABLE STAR U PEGASI. 
 
 15 
 
 of U Pcgasi's light curve near the maxima, with a view to ascertaining whether or not the 
 form of this curve near these epochs is real. 
 
 The appended plates will assist in forming a quick judgment of the degree of approxima- 
 tion of the theoretical to the observed curve. 
 
 Plate I. gives the points of Pickering's curve, the continuous curve drawn through these 
 points and used as the basis of the present discussion, together with the computed points 
 (marked with circles) of the theoretical curve. The position of the points of the derived 
 curve would conform much more closely to the curve of observations, by shifting the entire 
 observed curve before and after Min. II. forward by about 1.20 minutes, which, in view of 
 the short period of time over which the observations on which the constants of the equation 
 of the light changes depend, would be allowable. Plate II. shows the effect of this slight 
 shift. This, of course, amounts to assuming that Min. II. lies midway of the period, and 
 yet, since especial attention was directed to the study of the light change in this vicinity, 
 it does not seem that the difficulty would be likely to lie here. From private conversation 
 with Professor Pickering, I learn that the scarcity of observations at command and the 
 shortness of the interval over which his available observations were distributed made a definite 
 determination of the first constant of the equation of the light variation of this star impossible, 
 and that only the third decimal of a day can be relied on. This suggests the removal of the 
 difficulty by shifting the entire computed curve forward, or, what amounts to the same thing, 
 the entire theoretical curve backward by the above mentioned amount, and this gives a wholly 
 satisfactory accord of theory and observation for the entire curve save at the maxima. 
 
 Plate III. accordingly represents the observed curve in full line, the derived curve in dotted 
 line, and the latter, after the shift referred to, in a long dash followed by two shorter ones. The 
 computed points, enclosed in circles, are also given in their true (unshifted) positions. 
 
 Barring the vicinity of the maxima, for which further observations must be awaited, the rep- 
 resentation may, the writer thinks, be regarded as provisionally satisfactory, and that U Pegasi 
 is to be regarded as varying by reason of the mutual occultations of revolving components. 
 
 The following table contains for corresponding epochs the grade values of the ordinates 
 of both the computed and observed curves, together with the residuals in the sense observa- 
 tion computation for the final unshifted curve. Aside from the fact that the errors are 
 systematic, i. e., all of same sign (which are almost wholly gotten rid of by the aforesaid 
 shift), but little more could be desired, and the exceedingly small values of the average 
 deviations deprives the residuals of almost all significance. Applying the mean residual as 
 a correction to the individual residuals, which is the same as adopting the shifted curve as 
 final, the representation becomes entirely satisfactory. 
 
 COMPARISON OF COMPUTED WITH OBSERVED CURVE. 
 
 t 
 
 '. 
 
 >. 
 
 "_ 
 
 To 
 
 /. 
 
 "U 
 
 1.50 
 
 m. 
 
 9.35 
 
 m. 
 
 9.36 
 
 m. 
 
 -0.01 
 
 m. 
 
 9.34 
 
 m. 
 
 9.36 
 
 m. 
 
 -0.02 
 
 1.25 
 
 9.41 
 
 9.41 
 
 0.00 
 
 9.37 
 
 9.39 
 
 -0.02 
 
 1.00 
 
 9.48 
 
 9.48 
 
 0.00 
 
 9.43 
 
 9.45 
 
 -0.02 
 
 .75 
 
 9.58 
 
 9.58 
 
 0.00 
 
 9.52 
 
 9.52 
 
 0.00 
 
 .50 
 
 9.70 
 
 9.72 
 
 0.02 
 
 9.62 
 
 9.62 
 
 0.00 
 
 .25 
 
 9.84 
 
 9.87 
 
 0.03 
 
 9.72 
 
 9.73 
 
 -0.01 
 
 .00 
 
 9.90 
 
 9.90 
 
 0.00 
 
 9.75 
 
 9.75 
 
 0.00 
 
 Average Deviations = 0.009 = 0.01 
 
16 
 
 A STUDY OF THE LIGHT CURVE OF THE VARIABLE STAR U PEGASI. 
 
 Figure 4 illustrates the geometrical relations prevailing in the system, on the hypothesis 
 of separation of discs. The resemblance to /3 Lyrac is quite apparent, though there is an 
 essential difference in that, with the latter star, the smaller component is the brighter, while 
 with U Pegasi the reverse is the case. 
 
 FIG. 4. THE SYSTKM OF U PEGASI. 
 
 In conclusion, the writer would thank Dean Ricker of this University and Professor 
 Pickering of Harvard College Observatory for valuable assistance rendered during the prose- 
 cution of this inquiry : the former, by the loan of a computing machine, without which the 
 laborious computations involved in this paper could hardly have been made during the progress 
 of regular University work; and the latter, by granting the writer every possible means of 
 acquainting himself personally with the working methods and of forming an idea of the 
 attainable accuracy of the polarizing photometer. 
 
 CAMBRIDGE, MASS., August, 1808. 
 
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