r\A \ vi-n Lo ^ IY 9 l 
 
 THE 
 
 ASTROPHYSICAL JOURNAL 
 
 AN INTERNATIONAL REVIEW OF SPECTROSCOPY 
 AND ASTRONOMICAL. PHYSICS 
 
 JANUARY 
 
 NUMBER i 
 
 VOLUME VII 
 
 THE SYSTEM OF £ LYRAE. 
 
 By G. W. Myers. 
 
 Of the various theories which have been proposed to 
 explain the light changes of variables, three have been most 
 widely accepted : 
 
 First. A body whose surface possesses different degrees of 
 brightness^ different places, rotates upon an axis, bringing the 
 differently illuminated portions into view at regular intervals. 
 
 Second. A secondary meteoric swarm circles about a primary 
 swarm, so as to pass at regular intervals between the observer 
 - and the central swarm. The outlying meteors .of the two 
 " swarms, colliding with each other at the times of periastron 
 -"passage, produce also a periodic increase of brightness. A suit- 
 ^able combination of these two causes of light variation will 
 explain a large variety of the light fluctuations observed in varia- 
 
 Third. The so-called satellite theory, in which two bodies 
 whose luminous intensities may be either equal or unequal, 
 revolve about each other in orbits whose planes pass near the 
 solar system, and, bv the mutual eclipses of the bodies, light 
 changes of such character are produced as to explain some sort 
 of variability. 
 
 1 Read at the conferences held in connection with the dedication of the Yerkes 
 Observatory, Oct. 20, 1897. 
 
 P 
 
 lba<;5 
 
2 
 
 G. W. MYERS 
 
 The arbitrariness involved in the fundamental assumptions 
 of theories one and two, appears to me to be an element of 
 weakness of so serious a nature as to compel them to yield 
 to the third theory in all cases where the latter is applicable. 
 By proper assumptions, indeed, regarding the extent, mode 
 of distribution, and periodicity of the spots, or respecting the 
 periodicity and distribution of the meteors within the swarms, 
 as also the relative motions of the swarms themselves, any sort 
 of light change is capable of explanation by the first two theo¬ 
 ries ; but as has been pointed out, their well-nigh universal appli¬ 
 cability is in itself an element of weakness, inasmuch as the 
 theories in themselves are in no essential particular an advance 
 on their underlying hypotheses. 
 
 It is my purpose to give, in a few words, an account of a 
 recent attempt to represent the light changes of ft Lyrae, on the 
 basis of the so-called satellite theory. Argelander’s third curve 
 of this star was based upon I 500 careful photometric estimates, 
 extending over a period of nineteen years. An earlier curve by 
 Oudemans and a later by Schoenfeld, show no discrepancies 
 from this curve of a magnitude great enough to affect the dis¬ 
 cussion appreciably, and, consequently, Argelander’s third curve, 
 published in a pamphlet entitled, “ De Stella /3 Lyrae Variabili 
 Commentatio Altera” in 1859, was taken as the basis for the 
 discussion. This well-known curve is represented in the upper 
 portion of the accompanying figure. 
 
 INTRODUCTORY. 
 
 The variability of this star was discovered by Goodericke in 
 1784, but aside from the recognition of the fact of its variability 
 and the confirmation of the existence of two unequal minima by 
 this same scientist, nothing further was done in its study until 
 Argelander laid the foundation for a thoroughly scientific and, 
 at the same time, an extremely practical method of studying 
 the light fluctuations of variables. Applying his own method 
 to Lyrae, Argelander showed the star to have two unequal 
 minima, separated by two practically equal maxima, the entire 
 
SYSTEM OF /3 LYRAE 
 
 3 
 
 cycle of changes being completed in about I2 d 22 h ( = I2 d .gi). 
 At the time of the I Maximum, he found the brightness of the 
 star to rise to the 3.4 magnitude, and to fall after about three 
 
 days to a secondary minimum of the 3.9 magnitude, to rise again 
 after three days to the former brightness of the 3.4 magnitude 
 and then, after nearly the same interval, to fall to a primary 
 minimum of the 4.5 magnitude. Or, in other words, the bright¬ 
 ness at the maximum corresponds to 12.33 of Argelander’s 
 
4 
 
 G. W. MYERS 
 
 grades (0.127 mag.); at the primary minimum, to 3.34 grades 
 and at the secondary minimum, to 8.53 grades. 
 
 Assuming the light changes to be due to the mutual eclipses 
 of two revolving bodies of unequal brightness, we should have 
 the maxima occurring when the components stand beside each 
 other, without either being eclipsed by the other, and conse¬ 
 quently both disks shining full-phase. At the primary mini¬ 
 mum, the darker component lies in front of the brighter and 
 cuts off a portion of the latter’s light, while at the secondary 
 minimum the bright companion passes before the darker and 
 obscures a part of its inferior brilliance. The relative positions 
 of the components at the chief epochs are shown in the draw- 
 ing (Fig. 2). 
 
 
 Since the brightness of the “system ” is reduced at Minimum 
 I (primary minimum) by 66 per cent, of its greatest brightness 
 and at Minimum II (secondary minimum) by only 36 per cent., 
 it is evident (i),that the disks must be assumed unequally 
 bright or (2), that the orbital eccentricity must be assumed 
 great or finally, that both these circumstances concur. 
 
SYSTEM OF ft LYRAE 
 
 5 
 
 ECCENTRICITY. 
 
 An approximate idea of the magnitude of the eccentricity 
 may be obtained in the following two ways : 
 
 First. If we assume the motions of the components to be 
 in conformity to Kepler’s laws we have, for the relation of the 
 true and mean anomalies, the well-known equation : 
 
 v — M- f- 2 c. sin M- f- -f e 2 sin 2 M-f- .... 
 
 The lengths of the chief intervals of light change, from Arge- 
 lander’s curve are : 
 
 Min. I —Max. I —3.125 days. 
 Max. I —Min. II —3.250 i( 
 
 Min. II -— Max. II — 3-167 “ 
 
 Max. II — Min. I = 3.368 “ 
 
 The approximate equality of these intervals points unmis¬ 
 takably to a small orbital eccentricity. Assuming now, the 
 eccentricity to be so small as to render its higher powers negligi¬ 
 ble, we may shorten the above equation into : 
 
 (a) v = M-\- 2e sin M. 
 
 Designating the respective values of v and M at Min. I, Max. 
 I, Min. II and Max. II by v lf M l ; v 2 , M 2 ; v 3 , M 3 and v 4 , M 4 , 
 and substituting in (#), there result the following four equations : 
 
 (1) v x =M X -f 2<f. sin M 1 
 
 (2) v 2 — M 2 q- 2e. sin M 2 
 
 (3) = M 3 -(- 2c. sin M 3 
 
 (4) v 4 = M 4 + 2e. sin A / 4 
 
6 
 
 G. IV. MYERS 
 
 Subtracting (i) from (2), (3), and (4) in succession, and 
 
 . 
 
 noting 
 
 that v 2 - 
 
 ~ v \ — v i = 
 
 = 7 r ; and — - 
 
 obtain 
 
 
 
 
 (s) 
 
 7r 
 
 2 
 
 - 2 m x -)- 4 c. cos (M x 
 
 -f- vi x ) sin m x 
 
 (6) 
 
 7r 
 
 - 2m tl -\- /[e.cos (M x - }- 
 
 m 2 ) sin ;;/ 2 
 
 ( 7 ) 
 
 7r 
 
 3 2 
 
 - 2 m % -f- 4 e - cos (M x 
 
 -f- ^ 3 ) sin 
 
 where 
 
 2 VI j 
 
 = M^—M X = 3.125/x, 
 
 = 37° 8 ' .4 
 
 
 2 VI 
 
 = M S — M X 6.375/A 
 
 = 1 77 ° 46'.2 
 
 
 2 W 8 
 
 = M^ — M\ = 9.542/A 
 
 266° 4'. 8 
 
 and 
 
 fX 
 
 = 2tt T 5 , 7 J 
 
 12.91 days. 
 
 we 
 
 We shall then have but two unknowns in (5), (6) and (7), viz., 
 e. and M x , and any pair will suffice to determine these magni¬ 
 tudes. 
 
 From (5) and (6) we find, 
 
 (M j^i = -3°°28' 
 
 [ > | 0.0186, 
 
 and these values substituted in (1), give 
 v i=— 3 J ° 33 '- 
 
 This satisfies the intervals Max. I—Min. I, and Min. II—Min. I, 
 and requires Min. I to lie 31 0 .5 before periastron. 
 
 From (5) and (7), there result, similarly: 
 
 ( M x — 209°32'4 
 (y) -j e = 0.0196 
 
 l V x = 209 ° 24 ' . 
 
 Computing v x from Argelander’s curve, it will be found that 
 a rigorous satisfaction of the intervals Min. I to Max. I and Min. 1 
 to Min. II, requires a backward displacement of Argelander’s 
 Max. II by about 4 hours, while the intervals Min. I—Max. I 
 and Min. I—Max. II, used in getting (*:), necessitate a backward 
 displacement of Max. I by about 4 hours. Since now, so small 
 a shift in these two chief epochs, corresponds to so large a 
 
SYSTEM OF LYRAE 
 
 7 
 
 change (nearly i8o°) in the position of periastron, the eccen¬ 
 tricity of the orbit cannot be large. The mean of the two nearly 
 equal values of the eccentricity found above, is o.0191 ±0.0033. 
 This value is small enough to justify neglecting its second and 
 higher powers as was done above and thereby vindicates the 
 method of treatment. 
 
 Second. It will develop later that the hypothesis of a flatten¬ 
 ing of one or both bodies must be made. Assuming the bodies 
 to be deformed by reciprocal tidal influence, or by whatever 
 cause, into similar ellipsoids of revolution — a permissible assump¬ 
 tion, since such forms are figures of equilibrium-—and denoting 
 the ratio of the major and minor axes by “ q ,” so soon as an 
 approximate value of q is known, a superior limit of the ratio of 
 distance of centers, to the larger semi-major axis may be derived. 
 Thus, assuming the bodies to be spheres, with radii equal to 
 the respective semi-minor axes of the ellipsoids, a light curve 
 due to two revolving spheres may be computed. A lower limit 
 for the duration of the eclipse at Min. I, for example, may be 
 read off from this curve. A little reflection will show that the 
 larger q be taken, the shorter the duration of the eclipse will be. 
 Taking then, a value of q greater than that tound later to be 
 the approximate value and assuming that the eclipse has not 
 begun until the light curve has fallen considerably, a value for 
 the eclipse-duration will be obtained which is, at all events, 
 small enough, perhaps much too small, q is later found to be 
 1.2, and assuming it to be 1.3, I find for inferior limit of eclipse- 
 duration, 3 days and 4 hours, which must be at all events, small 
 enough. But the smaller this inferior limit, the larger must be 
 the ratio of distance of centers to the larger semi-major axis. 
 Using the above value 3 d 4 h for eclipse-duration, a superior limit 
 for this ratio may then be obtained, which will at any rate, be 
 large enough. Taking now, the larger radius as unity and 
 denoting the smaller by *, the radius vector of the true orbit 
 by r (for this, e may be put - o), one half the distance between 
 the nearest points of the satellite at the beginning and end of 
 the eclipse, by we see from the figure that, 
 
8 
 
 G. IV. MYERS 
 
 CPC' ^3. 167X27°. 887 = 88° 20' 
 and hence CPD >44°-10'. 
 
 Also r V (jv T k) c s c 44 0 io' 
 
 1(X + K) I. 438 . 
 
 But since x^i , I, we haver<2.87 times the smaller 
 semi-axis of the larger ellipsoid or = 2.4 times its larger semi¬ 
 axis. The distance between centers then, being so small com¬ 
 pared with the dimensions of the larger body, the eccentricity 
 could not be large or the masses would necessarily interpene¬ 
 trate during revolution. 
 
 These considerations are sufficient to warrant the assump¬ 
 tion of a small orbital eccentricity and to justify the hypothesis, 
 that a first approximation to the orbit may be obtained by mak¬ 
 ing e = 0. 
 
 FLATTENING. 
 
 The necessity of the assumption either that the disks are 
 flattened, or that the bodies are not yet separated, is apparent 
 from the fact that the brightness at the maxima, does not remain 
 constant for any considerable length of time. It has been 
 assumed in what follows that : 
 
 (1) The two bodies are distinct and separate. 
 
 (2) Both are deformed into ellipsoids of revolution. 
 
 (3) The periods of rotation and of revolution are equal to 
 I2 d .9I. 
 
 (Possible librations being disregarded.) 
 
SYSTEM OF /3 LYRAE 
 
 9 
 
 Taking the origin of coordinates at the center of the larger 
 ellipsoid and assuming the brightness of the disks uniform, we 
 may readily find from the principles of analytical geometry of 
 space, the equation of the light curve, which may be used while 
 the bodies stand wholly off each other. It was found to be 
 
 X 
 
 i + A j | / s i n 2 <£ + (R/R'y cos 2 <j> 
 
 + — = l 
 
 I sin 2 <£ + (B/B'Y cos 2 (j> \ 
 
 Where L denotes the ratio of the combined luminosities of 
 both simultaneous elliptical disks to the combined brightness 
 when the areas of these disks are greatest, i. e. y at the instant 
 of the maxima : 
 
 A. is the ratio of the intensities of the disks, 
 
 k is the ratio of the corresponding semi-axes of the two ellipsoids. 
 
 B ', R' and B, R denote respectively the semi-major and semi¬ 
 minor axes of the smaller and larger ellipsoids, 6bcisthe sight line and 
 ZOy the tangent plane to the celestial sphere at the system. 
 
o 
 
 G. W. MYERS 
 
 cf> is the longitude of the secondary in its orbit. 
 
 As a first approximation q was put equal to R'/R = B'/ B. 
 
 From the unsymmetrical character and small magnitude of the 
 final residuals furnished by a comparison of Argelander’s curve 
 with the computed curve, it appears later, that nothing can be 
 gained by an attempt to improve this hypothesis by ascribing 
 different degrees of flattening to the two ellipsoids, so long as 
 both bodies be regarded symmetrical and their disks uniformly 
 bright. On this hypothesis the above equation becomes 
 
 i i 
 
 Z= 7 0 _ . , where cf> — -—- t U being in hours). 
 
 V sin 2 cf> + i / q 2 cos 2 12 P v 6 ’ 
 
 An approximate value of q was found by computing light 
 curves for various values of q , viz., i.i; 1.2; 1.25; and 1.3, 
 through a number of symmetrically chosen points before and 
 after the maxima, and deducting the computed values of the 
 ordinates from those of Argelander’s curve. The correct value 
 of q should, of course, give a curve whose ordinates, deducted 
 from the corresponding ordinates of Argelander’s curve, would 
 be those of a curve due to two revolving spheres, and for a time 
 on either side of the maxima, i.e ., while the two components are 
 wholly uncovered, such a curve must run horizontally. The 
 value 1.2 of q gave the following nearly equal ordinates : 
 
 10.33; 10.55; 10.76; 10.78; 10.78; 10.79; 10.78; 10.78; 10.76; 
 10.55 ; and IO -33- 
 
 This value of q was then assumed as a first approximation. 
 
 An equation of a light curve applicable during the eclipses 
 was then computed by the process suggested in the drawing, 
 and for Min. I, the necessary equations were found to be : 
 
 J‘ ' - .77“+ k- a, j * + (»-*')- P‘’ si® * [ 
 
 , p sin (cb±<b') , .. o . „ . o 
 
 pr . 7 ; according as <f> < 90 or <f> >> 90 , 
 
SYSTEM OF /3 LYRAE 
 
 I I 
 
 Putting M,= tt(i + k 2 X) (i—//). 
 
 Then j- k 2 <£" — p' sin <j> or J/ 7 — -|- * 2 — #c sin(<£" + <£). 
 
 The work of computation was somewhat shortened by intro¬ 
 ducing an additional auxiliary H h when <£">^ and by using the 
 
 foregoing equations only when (/>"<-. Hj was defined by 
 Hj= K 2 7T— 7T (i -f K 2 A.) (i-//), 
 
 Whereupon 
 
 Hj—k 1 <£'-*—<£—p' sin <£, or H 1 = k 2 <£'— </> -j- k sin (<j>' — <£) 
 and similarly, the equations for Min. II were found to be : 
 
 ^"' =I_ ,r(i+,c* A) { <A + k ' 2 (*- + ')-p'sta * ( 
 
 1 1 ■ K 2 A. i / 
 
 M fI =ir ——- j 1 — Jn , or — -f- k 2 <f) lf — p' sin <f> 
 
 H u = k 2 tt — 7T- 1 —- (i—///'), or K 2 <£"— <£-fV sin <£. 
 
 A 
 
 The remaining formulae hold without modification for both 
 minima. Jj' and J If r are the ratios of the combined instantaneous 
 brightness to the combined brightness due to the sum of the 
 two instantaneous disks shining full-phase during the eclipses at 
 Min. I and Min. II respectively. 
 
G, W. MYERS 
 
 I 2 
 
 J r and J ir are the ratios of the same brightness to that due 
 to the sum of the full disks, when these disks are of maximum 
 area. 
 
 The relation between J/ and / 7 on the one hand, and J n ' 
 and J If on the other, were then derived and found to be 
 
 Ji=fJ/ and Jn=fJ n ' , where f= R/ /R = B/ /B 
 
 i 
 
 j '"sin 2 /?-f- ^ 2 cos 2 /?’ 
 
 and /3 cf) — - denotes the longitude in the orbit, counted from 
 
 a point 90° ahead of the origin of (f>. (For undefined symbols 
 see Fig. 6.) 
 
 Designating by / 3 -|- ^ , the true anomaly in the real orbit, and 
 
 by a that in the apparent orbit, both counted from the node, and 
 calling p and r the radii vectores in the apparent and true orbits 
 respectively, we have 
 
 r. sin (3 = p cos a and tan a = cos i cot / 3 , 
 wherefore p‘ 2 =r 2 . sin 2 fi-\-r 2 . cos 2 i. cos ( 3 . 
 
 Differentiating the formulae for Min. I and reducing, we find 
 8M f SB, 
 
 8 <f> 
 
 and 8 </> 
 
 2 k tg (f)" sin ((f)” -f- (f)) ~~ ^ 2 Ktg <f)' sin (<f)' — <£)’ 
 
 and when tcy> 1, as will be found to be the case later, the pre¬ 
 ceding equations become 
 
 Mj= (f> -j - k 2 (f> 1 — k sin (<f) -j- ^') 
 
 H, — (f) — k 2 (f>' T k sin ( 4 > — (/>') 
 
 , « sin ((f) — (f)') 
 
 p 
 
 sin (f) 
 
 8c/> — 
 
 BET, 
 
 2 k tg (f)' sin ((f)-- (f >' ) 
 
 / was obtained by subtracting the ordinate of Argelander’s 
 curve for the selected instants from the mean of the maximum 
 ordinates, calling this difference A G, and computing J from the 
 equation log /= 0.051 A G , which is readily derived from Pog- 
 sjii’s scale, together with the value of Argelander’s grade. The 
 
SYSTEM OF (3 LYRAE 
 
 3 
 
 values of M Iy M in H f , H IV are computed directly from values of 
 J obtained from the above curve, and then an approximate value 
 of <£, interpolated from tables computed from the M's and H’s 
 with (f) as an argument, are then corrected by the above differen¬ 
 tial formulae. Thus it is possible by these formulae to compute a 
 light curve from known elements, and also to solve the converse 
 problem. To compute these tables the values of k, were required. 
 k being unknown, an approximation of its value was obtained 
 thus : calling b m the brightness of a star of magnitude m , and 
 b M that of a star of magnitude M, by Pogson’s scale 
 
 log = log / 0.4 A M, 
 
 where £s,M i s the difference of the brightness in stellar magni¬ 
 tudes. From this we have 
 
 Brightness at Min. II 
 Brightness at Min. I 
 
 1-8536 
 
 Brightness at Max. 
 Brightness at Min. I 
 
 The first of these gives 
 
 (<0 
 
 and the second 
 
 w 
 
 I + K 2 A — a k 2 A 
 
 I —a K 2 -j- K 2 A 
 
 I —j— a * 2 + K 2 A 
 I — a k 2 -j- k 2 A 
 
 m, 
 
 where a denotes the portion of the disks common to both bodies 
 at the middle of the eclipses. 
 
 By a few simple transformations of these equations, it may 
 be readily seen that 
 
 q > 1.0205, *• e - the disks must be flattened, and 0.8323 1.5241. 
 
 The values of /c selected, were then 
 
 0.8323, 0.9049, 1.0000, 1.2883 an d *-5241 
 
 and a table such as was mentioned above was computed with the 
 argument <£ or <£' according as k< i, or k> i. 
 
G. W. MYERS 
 
 14 
 
 The values of r y for various points before and after the min¬ 
 ima, with a correct hypothesis for re and i, the inclination, must 
 all be approximately equal, since they are computed on the 
 hypothesis of a circular orbit, k- 1.5241 gave a fair approach 
 to an agreement in the various values, while the other values of 
 /c gave widely discordant results for r. For some reasons it was 
 more convenient to have k< i, and hence, the larger radius was 
 assumed unity and the necessary alterations in the formulae were 
 made to suit this hypothesis. The only possible assumption 
 
 which could be made for i was shown to be -, since this gave 
 
 the various values of r more nearly equal than any other assump¬ 
 tion for it. A was found to be 0.3353 from ( d ) and (^). 
 Recapitulating then; the following values were taken as first 
 approximations : 
 
 e - -- o, i= -, r 1.8344, k 0.6 q61, A ~ 0.3353 ar| d <7=1.2. 
 
 2 
 
 By differentiating the above formulae and combining the 
 results, the following differential equations for correcting the 
 approximate circular elements were derived : 
 
 (I )*(\ + *)JJ,= 2KK l d*+ J " Sin *' dr + r* C0S,gsin *' d (.■'») 
 
 r o 
 
 d' 
 
 •0» 
 
 where K f = A J t f 2 cos 2 —/,)—/<£— Cpf 2 cos 2 (3 sin 
 
 and 
 
 A — Trq\, J3 - tt ^ 1 + ^ ^ k2 — 
 
 C- 
 
 2 ?3 ‘ 
 
 m 
 
 q must not be included in this equation since it depends on k 
 by equation (e). The equation applies only during Min. I. For 
 Min. II, 
 
 / t T \ tt(A+k 2 ) 7r ^ 7 , 2 p sin <f>' , r 2 cos 2 /? sin <j>' , 
 
 (II) —^- dJ//=2K K n dK -|- r - dr-\ - d (i ) 2 
 
 A r p 
 
 K„= A J " f - x — P + tt (/—//,)— Cp/ 2 cos 2 p sin . 
 
SYSTEM OF p LYRAE 
 
 15 
 
 The coefficients of (I) and (II) were computed for 14 points 
 selected symmetrically along the curve, with the approximate 
 values given above, dj, and dj n were obtained by comparing 
 corresponding ordinates of Argelander’s curve with those of a 
 light curve computed from the preceding approximate values. 
 14 observation equations lead to the following values: 
 
 dK — -f 0.3586, dr— -j- 0.0999, and d(i' z ) = —0.0434 
 
 i’ being here imaginary, but numerically small, it was called 0, 
 and dK being quite large, dq was introduced in its stead, since 
 small changes in the fundamental data do not affect q so greatly 
 as r, and the equations again solved, gave the corrections, 
 dq ~= — 0.0007, an d dr= —0.0889, and the corrected values 
 were then <71.1993 and r 1.8955, and from ( d ) and ( e ), 
 k 0.7580 and X —0.4023. The probable error was, of course, 
 somewhat increased by dropping V , being in the latter case 
 ±0.1 of a “brightness.” These values were regarded as the 
 most probable on the assumption of a circular orbit. 
 
 By means of these values and differential equations, which 
 were derived for correcting a circular into an elliptical orbit of 
 small eccentricity, a set of 48 observation equations was obtained, 
 for as many points of the light curve, and from them the follow¬ 
 ing elliptical elements were obtained : 
 
 T 1855 Jan 13 d 6\ 35 
 
 P 
 
 12.91 d 
 
 0 =i -937 
 
 
 = 27°.887 
 
 i =90 0 
 
 K - 
 
 = 0.7528 
 
 rj 94 0 53' from node 
 
 A 
 
 o -399 
 
 0=-o° (assumed) 
 
 q- 
 
 = 1.203 
 
 t = 6 d i5 h -4 after Min. I. 
 
 Both e and 1 were included in the equation and i' was again 
 imaginary but very small. 
 
 Using the equations derived for computing the light curve 
 for elliptical motion, the results of computation are comprised 
 in the columns of the table given below. The columns headed 
 A andy B contain the computed and observed ordinates respec¬ 
 tively, and those headed A/ B _ R , the corresponding residuals 
 
G. W. MYERS 
 
 16 
 
 from Argelander’s curve. The computed curve is shown in 
 (Fig. i rt ) drawn in a dotted line beside Argelander’s curve 
 drawn in a full.line. The agreement is quite close. 
 
 TABLE OF COMPUTED AND OBSERVED BRIGHTNESS. 
 
 
 
 Minimum I 
 
 
 Minimum II 
 
 
 •A 
 
 
 5/ b-r 
 
 
 
 S-'B-R 
 
 —72 
 
 0.9963 
 
 O.9956 
 
 ' +0.0007 
 
 O.9918 
 
 O.9970 
 
 —0.0052 
 
 —66 
 
 0.9870 
 
 O.9852 
 
 +0.0018 
 
 O.9836 
 
 0-9945 
 
 —0.0109 
 
 —60 
 
 0.9683 
 
 0.9701 
 
 —0.0018 
 
 O.9696 
 
 0.9833 
 
 —0.0137 
 
 —54 
 
 0.9524 
 
 0-9513 
 
 +0.001 I 
 
 O.949O 
 
 O.9672 
 
 —0.0182 
 
 -48 
 
 0.9147 
 
 O.9270 
 
 —0.0123 
 
 O.9258 
 
 O.9482 
 
 —0.0224 
 
 —42 
 
 0.8732 
 
 O.8849 
 
 -0.0117 
 
 0.8995 
 
 O.9217 
 
 —0.0222 
 
 —36 
 
 0.8209 
 
 0.8268 
 
 —0.0059 
 
 0.8680 
 
 0.8886 
 
 —0.0206 
 
 — 3 ° 
 
 0.7296 
 
 0.7525 
 
 —0.0229 
 
 O.8306 
 
 0.8494 
 
 —0.0188 
 
 —24 
 
 0.5836 
 
 O.6CI9 
 
 —0.0183 
 
 O.7836 
 
 0.8048 
 
 —0.0212 
 
 — 18 
 
 0-4336 
 
 0.4993 
 
 —0.0657 
 
 O.7246 
 
 0.7558 
 
 —0.0312 
 
 —12 
 
 0.3661 
 
 0.4275 
 
 —0.0614 
 
 O.6674 
 
 0.7044 
 
 — 0.0370 
 
 - 6 
 
 0.3484 
 
 O.3487 
 
 —0.0003 
 
 O.6433 
 
 0.6542 
 
 —0.0109 
 
 0 
 
 0-3433 
 
 0-3433 
 
 +0 
 
 O.6365 
 
 0.6368 
 
 —0.0003 
 
 + 6 
 
 0.3499 
 
 O.3488 
 
 +0.0011 
 
 O.6472 
 
 0.6372 
 
 + 0.0100 
 
 +12 
 
 0.3988 
 
 0.4275 
 
 —0.0287 
 
 O.6762 
 
 0.6689 
 
 + 0.0073 
 
 +18 
 
 0.5306 
 
 0.5591 
 
 —0.0285 
 
 0.7340 
 
 0.7203 
 
 +O.OI 37 
 
 +24 
 
 0.6572 
 
 O.6624 
 
 +0.0052 
 
 O.7978 
 
 0.7716 
 
 +0.0262 
 
 +30 
 
 0.7644 
 
 O.7528 
 
 +0.0116 
 
 O.8498 
 
 0.8289 
 
 +O.O2O9 
 
 +36 
 
 0.8416 
 
 O.8266 
 
 +0.0150 
 
 O.893I 
 
 0.8622 
 
 +O.O3O9 
 
 +42 
 
 0.8857 
 
 O.8845 
 
 +0.0012 
 
 0.9255 
 
 0.8997 
 
 +0.0258 
 
 +48 
 
 0.9234 
 
 O.9268 
 
 —0.0034 
 
 0.9524 
 
 0.9307 
 
 +0.0217 
 
 +54 
 
 0-9537 
 
 O.9508 
 
 +0.0029 
 
 O.9627 
 
 0-9545 
 
 + 0.0082 
 
 +60 
 
 0.9732 
 
 O.9694 
 
 +0.0038 
 
 O.9881 
 
 0.9732 
 
 +O.OI49 
 
 + 66 
 
 0.9870 
 
 O.9847 
 
 +0.0023 
 
 0.9977 
 
 0.9877 
 
 +0.0100 
 
 +72 
 
 0.9940 
 
 0.9952 
 
 —0.0012 
 
 I.OO49 
 
 0.9970 
 
 +0.0079 
 
 With the help of an eccentricity, therefore, the residuals are 
 somewhat reduced, and, considering the errors necessarily attach¬ 
 ing to photometric estimates, the curve of Argelander may be 
 regarded as sufficiently well represented. 
 
 The eccentricity results again almost the same as before, so 
 that at the epoch 1855, the eccentricity did not differ materially 
 from 0.02. 
 
 Since the periastron lies near Min II, the systematic devia¬ 
 tion of the computed from Argelander’s curve, the former lying 
 above the latter, before, and below it, after Min II, it is highly 
 probable, that while the secondary rounds periastron, very con- 
 
SYSTEM OF /3 LYRAE 
 
 '7 
 
 siderable augmentation of brightness due to deformations of the 
 disks, to internal friction, etc., occurs. Because of inertia, these 
 effects could not immediately show themselves, so that the real 
 curve would lie below the computed mean before Min II and 
 above it after Min II, as is represented in the figure. 
 
 Feeling somewhat suspicious that any one of several sets of 
 elements lying near those just given might represent observa¬ 
 tions equally well, it seemed worth while to test whether this 
 same set of elements would result, if one of the circular elements 
 upon which the elliptical ones were based, were given an arbi¬ 
 trary change and then, by repeated applications of the method 
 of least squares, the elements were again corrected by the obser¬ 
 vations. An arbitrary change of —0.01 was given to k. and of 
 — 0.07 to r and after four adjustments giving ever smaller proba¬ 
 ble errors, the former values were again obtained. 
 
 It may therefore be inferred that the above elements are the 
 most probable. An interesting fact arising during the latter 
 process of adjustment, was that one set of elements giving a 
 probable error of nearly the same magnitude as the final set, 
 gave the distance between the centers =fe§ 1.80 and the sum of the 
 radii equal to 1.82, i. e. } the components are not yet separated. 
 This fact in connection with the low mean density of the system 
 points to the nebulous condition of the star. The indications are 
 then, either that the companions are not yet separate, but in the 
 act of separation, or that if separate, their separation has taken 
 place comparatively recently. In either case we seem to have 
 here the first concrete example of a world in the act of being 
 born. 
 
 SPECTROSCOPIC CONSIDERATIONS. 
 
 Treating by the method of Rambaut, published in Mon. Not. 
 51, 316, the observations of Belopolsky made between Sept. 23 
 and Nov. 26, 1892, and published in Melanges math, et astrono- 
 miques , t. VII, l. 3, I find, 
 
 V = — 0.8 kilometers per second 
 T— 26.93 September 1892 
 
G. IV. MYERS 
 
 I 8 
 
 e = 0.108 
 
 o> —79 0 17' from node 
 a sin i — 15836000 kilometers 
 
 P = i2 d .9i from light period. 
 
 Lockyer observes the relative displacements of three lines 
 and finds velocities as follows : 
 
 Hy — 155.0 miles 
 
 178 = 154-0 miles 
 
 ^4025= i5 8 -° mdes. 
 
 The mean of these values is 155.7 miles and from a private 
 letter of Professor Lockyer, 1 find it belongs to the epoch, 
 August 24.46, 1893. Correcting for the motion of the Earth and 
 reducing to kilometers, there results for the relative velocity in 
 the line of sight 259.8 kilometers. 
 
 Belopolsky’s measures give for the diameter of the absolute 
 orbit of one of the components 31672000 kilometers. From 
 the absolute orbit furnished by Belopolsky’s observations, the 
 velocity for the above date was found to be of the above 
 
 relative velocity. The semi-major axis of the relative orbit is 
 then A = 3.168/2X 31672000 — 50175000 kilometers (calling 
 
 7 T 
 
 i =~ - ) and for the ratio of the masses 
 
 2 
 
 a ' A = m M), or m /M = -— • 
 
 Assuming that the F-line observed by Belopolsky was pro¬ 
 duced by the smaller component, we have 
 
 A /a = (Mm ) /m = 1 -|- 8 ' k 3 , 
 
 (S' —ratio of densities of components), 
 and assuming his line to be produced by the larger, there results, 
 A /a = (M- f- m) /m 1 -j- 1 S'k 3 . 
 
 Using the values of a/A and k given above, we find 
 8 '= 5.083, or 1.081. 
 
 This furnishes a means of deciding which of the components 
 produced Belopolsky’s F-line. Since it is quite improbable that 
 one of the two bodies, so near to each other as is the case here, 
 should have a density 5.083 times that of the other, the latter 
 
SYSTEM OF ft LYRAE 
 
 19 
 
 value is assumed to be the correct one, and hence, that Belopol¬ 
 sky’s F-line was due to the larger component. 
 
 Using now the well known relation 
 
 M+m = A* /F* 
 
 and substituting the values of A in astronomical units and of 
 P in years, there result, 
 
 M- f- m — 30.56 solar masses 
 and since m/M= 1 / 2.168, 
 
 M= 20.91 solar masses 
 and m — 9.65 solar masses. 
 
 Calling 5 the solar mass, H the solar radius, R' and R, the 
 semi-major and semi-minor axes of the minor ellipsoid, the 
 density of the larger companion in terms of the solar density 
 and S 2 of the smaller, 
 
 Sj = (M/S) q* (H /R')z, (q~R' R as before) 
 
 and substituting former values, 
 
 = 0.00058. 
 
 But 
 
 8 2 = 1.08 Sj (8' = 1.08) 
 
 , hence 
 
 8 2 — 0.00063. 
 
 The mean density of the system is then somewhat less than 
 the density of air, i. e ., comparable to nebular density. It 
 appears then that (3 Lyrae furnishes us a concrete illustration of 
 the actual existence in space of a Poincare figure of equilibrium. 
 
 Using the chief epochs of Lindemann’s curve constructed 
 from Plassmann’s observations, the following equations result: 
 
 cos (M x + 49° 57') = —0.0563/^ = — 3*204 (with Argelander’s e) 
 cos (M t 4 - 93 0 5 2 ') = — 0.0338/V = — 1.920 
 cos ( M 1 -)- 136° 23'.5) = — 0.01771 /e — — 1.010. 
 
 The impossibility of these relations, on the hypothesis that e 
 is not greater than it was in 1855, requires us to infer that since 
 Argelander’s time the eccentricity of the system has grown 
 larger. Comparing the spectroscopic with the photometric 
 
20 
 
 G. IV. MYERS 
 
 observations, it is also seen that a large motion of the line of 
 apsides has occurred since 1855, though its precise amount 
 can scarcely be ascertained with any considerable degree of 
 certainty. 
 
 The following results, therefore, seem to be quite clearly 
 indicated by the preceding discussion : 
 
 1. The photometric estimates of /3 Lyrae’s variability may be 
 explained easily within the limits of the errors of these esti¬ 
 mates by the aid of the satellite theory. 
 
 2. The bodies may be regarded as similar ellipsoids of revo¬ 
 lution. 
 
 3. The orbit of the secondary body is nearly circular and its 
 plane passes almost exactly through the Sun. 
 
 4. The common flattening of the ellipsoids differs but little 
 from 0.17 and, aside from librations, the periods of rotation and 
 revolution are equal. 
 
 5. The larger body is about 0.4 as bright as the smaller. 
 
 6. The distance of centers is extremely small, about \ l/% of 
 the semi-major axis of the larger ellipsoid. 
 
 7. From Lindemann’s chief epochs for 1892 the orbital 
 eccentricity of the system must have increased from 1855. 
 Belopolsky’s spectroscopic observations indicate the same. 
 
 8. The motion of the center of gravity of the system with 
 respect to the Sun is very small. 
 
 9. The semi-major axis of the orbit of the companion is 
 about 50,000,000 kilos. 
 
 10. The mass of the larger body is 21 times, and of the 
 smaller 9.5 times the solar mass. 
 
 11. The densities of the companions are nearly the same. 
 
 12. The mean density of the system is comparable with 
 atmospheric density, or the “system” (for such, I think, it 
 must now be called), is in a nebulous condition. 
 
 13. In conclusion, it may be said that the spectroscopic 
 and photometric observations, which were available to me for 
 the foregoing discussion, so far from being widely discordant, 
 as some have thought, agree with each other remarkably closely. 
 
SYSTEM OF LYRAE 
 
 21 
 
 The strong absorption lines in the spectrum of this star point 
 to the presence of a powerfully absorbing atmospheric envelope 
 about the nuclei of the masses. From the dynamical theory of 
 gases we know that such an atmospheric layer would arrange 
 itself about the combined mass of the system so that portions 
 of equal density would be in equipotential surfaces. These 
 surfaces would, in the immediate vicinity of the surfaces of the 
 masses, conform somewhat closely to the surface of the bodies, 
 but they would rapidly lose the abrupt curvatures at the surface 
 and become more and more nearly spherical. A rough attempt 
 to represent this is shown in the subjoined figure. 
 
 The equipotential surfaces, shown by dotted lines, would 
 dispose themselves svmmetrically about the center of gravity 
 of the system as a center, and if the density of the larger com¬ 
 panion were very materially greater than that of the smaller, 
 the center of gravity would lie far within the larger compan¬ 
 ion. It might even happen that the center of gravity of the 
 two bodies should lie beyond the geometrical center of the 
 larger ellipsoid. Where it would lie would depend wholly upon 
 the density of the various parts of the mass of the two bodies. 
 It is then readily seen that the atmosphere might be, and prob¬ 
 ably is, so arranged as to permit the remote end of the smaller 
 
22 
 
 G. W. MYERS 
 
 to shine through a shallow layer of it and thereby to permit 
 the smaller to appear brighter, even though it might be intrin¬ 
 sically darker than the larger body. No violence is done to 
 theory, at all events, by such an assumption. This distribution 
 of the atmosphere would also explain the absorption bands, 
 which are seen in the spectrum of this star more distinctly at 
 Min I than at Min II. The continuous spectrum, which would 
 be produced most distinctly by the smaller body, must at the 
 same time appear fainter. This accords with spectroscopic 
 observations also. 
 
 Although some of the ideas given above may seem a little 
 venturesome, let it be remembered that the peculiar character 
 of the observations of this star leads one to expect an explana¬ 
 tion of a somewhat, unusual nature. 
 
 That the ellipsoids are similar is, of course, an arbitrary 
 assumption. 
 
 In conclusion, let it be observed that an attempt at a formal 
 representation of the condition of things prevailing in the sys¬ 
 tem of ft Lyrae, leads to the assumption of a single body (such 
 as Poincare’s or Darwin’s figures of equilibrium). The above 
 has, of course, only a formal significance, but on account of the 
 poverty of observational material at my disposal an attempt to 
 push the discussion farther on a mathematical basis could not 
 have proved profitable. It is believed, however, that the discus¬ 
 sion may help us to orient our views with regard to this wonder¬ 
 fully interesting star. Fig. i b represents the most probable 
 relative dimensions of the bodies and orbit of the system, as 
 based on Argelander’s photometric estimates up to 1859. Pro¬ 
 fessor Pickering has kindly offered to place all the earlier esti¬ 
 mates of /3 Lyrae’s brightness at the writer’s disposal, and it is 
 the latter’s intention to make a full investigation and discussion 
 of them at the earliest possible date.