a\9 ■^ — " c" o« I ■ i^ ? •/? r^ S J T' ^"l' LJ ■o cr '4" '=^ r c3 rf 1 (Sioxmll Unimieiitg pilrtajg THE GIFT OF .A2.CUvwvjL.cyut. -XSvsAiXUAwzvs.. ..k.\y:3.H.5a 3./n:?>-H.. H87 Cornell University Library QB 819.N53 On the position of tlie galactic and othe 3 1924 012 310 326 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012310326 CONTRIBUTIONS TO STELLAR STATISTICS FIRST PAPER ON THE POSITION OF THE GALACTIC AND OTHER PRINCIPAL PLANES TOWARD WHICH THE STARS TEND TO CROWD BY SIMON NEWCOMB WASHINGTON, U. S. A.: PUBLISHED BY THE CARNEGIE INSTITUTION June, 1904 CONTRIBUTIONS TO STELLAR STATISTICS FIRST PAPER ON THE POSITION OF THE GALACTIC AND OTHER PRINCIPAL PLANES TOWARD WHICH THE STARS TEND TO CROWD BY SIMON NEWCOMB WASHINGTON, U. S. A.: PUBLISHED BY THE CARNEGIE INSTITUTION June, 1904 CARNEGIE INSTITUTION OF WASHINGTON Publication No. io Press of The New Era pHtNTiNG Cohpani^ Lancaster. PA ON THE POSITION OF THE GALACTIC AND OTHER PRINCIPAL PLANES TOWARD WHICH THE STARS TEND TO CROWD. FAGB. PAGE § I. The problem Stated i § 7. Properties of the principal planes 11 2. The plane of condensation 3 8. Weight with which the plane of condensation 3. Relations of the three planes given by the solu- is determined 12 t'°° 4 9. Solutions of the equations 14 4. Properties of the planes S 10. Position of the central plane of the galaxy 16 5. The roots of the fundamental cubic 6 11. Gould's belt 18 6. Professor Moore's generalized demonstration of 12. Principal planes of stars of various classes 20 the invariance of the roots of the cubic under 13. Law of richness of the galactic region 24 an orthogonal transformation 9 14. The problem of totality of star-light 29 § I. The Problem Stated. It is well known that the sky appears to us poorest in stars in the regions around the poles of the galax}?^, and that it continually grows richer at a rate which is slow at first, but more rapid afterwards, from the poles toward the galactic circle. This law of increase forms the basis of Herschel's theory of the figure of the stellar system, and led him to the conclusion that the latter extended much farther in the direction around the galaxy than in the directions of the galactic poles. ' An examination of the features presented by the galaxy shows, however, that we have here to do with something more than a general condensation of the stars toward its central circle. In the equality of the number of stars around the two poles of the Milky Way, and in the regularity with which the numbers increase toward the galactic circle, we find reason to believe that the thickness of the stars in space is, at least for the region of the system within the galactic girdle, approxi- mately constant. But the appearance of the galaxy is irreconcilable with any hypothesis of equal thickness. The most obvious to vision of its features is the Note. — The completion of this paper was made possible by a grant from the Trustees of the Gould fund, which enabled the author to avail himself of the valuable services of Mr. W. T. Carrigan in performing the more laborious of the required computations. 2 PRINCIPAL PLANES OF THE STARS. collection of the stars which compose it into agglomerations which- frequently have fairly well defined boundaries. To explain these seeming agglomerations by a col- lection of stars of equal thickness we should have to suppose the galaxy formed of groups of stars having the form of long narrow cylinders, a form too fantastic to merit consideration. The only rational hypothesis we can form is that these agglomera- tions are real, and that, within them, the stars are much thicker than elsewhere. We are thus led to make a distinction between the stars that form the Milky Way and other stars. This distinction is especially accentuated in the region extending from Cygnus through Aquila to the Southern Cross, where the Milky Way is marked by a number of irregular openings interspersed among comparatively brilliant agglom- erations of stars. The stars which form the Milky Way are also distinguished by the large propor- tion of those whose spectra show a predominance of blue and therefore belong to the first type. That galactic condensations are formed entirely of stars of this type cannot be asserted ; but it is quite likely that, when their spectra are cata- logued, we may find them to afford great assistance in separating the galactic from the non-galactic stars. One object of the present paper is to determine the position both of the galaxy itself and of the planes toward which the stars appear to crowd, irrespective of the existence of the galaxy. What can be done in this direction is suggested by the planispheres on which Schiaparelli laid down the density of each portion of the sky in lucid stars. These show that by drawing a great circle around the sphere, through the regions richest in lucid stars, we shall reproduce the position of the galaxy within a few degrees. It might therefore seem that the tendency to crowd toward the galaxy is well marked even in the case of the stars visible to the naked eye. We shall, however, see that this tendency is less marked when we consider the galactic stars as forming a collection separate from the others. The problem as worked out in the present paper makes no hypothesis as to the actual thickness of the stars in space, but considers only their apparent distribution in the sky. We may, therefore, for our immediate purpose, consider all the stars as if at an equal distance, which we take for unity. I then define their plane of condensation in the following way: Let us suppose a plane taken at pleasure pass- ing through our position in the universe, which point we take as the origin of co-ordinates. This plane will cut the celestial sphere in a great circle. The per- pendicular distance of a star from the plane will then be represented by the sine of its distance from the great circle. Let us form the sum of the squares of these sines for the whole system of stars which we consider. The value of this sum will vary with the position which we assign to the plane. The principal plane of condensation, as I define it, is that for which the sum in question is a minimum. plane op condensation. 3 § 2. The Plane of Condensation. To investigate this plane, or to find the great circle in which it intersects the celestial sphere, let us put: a, by c, the co-ordinates of any one of the given stars referred to any system of rectangular axes. We take for axes in the first place, as usual, the lines passing through the celes- tial pole, the equinox, and the point on the equator in 90° of R. A. We shall then have, in the case of each star of given Right Ascension and Declination, a and 8; a = cos S cos a b = cos S sin a c— sin h These three quantities are the cosines of the angular distances of the star from the three points above described. Let us also put: X, Y, z, the cosines of the angular distances of the pole of the required great circle from the same three points. p, the distance of the star (at, b, c) from this plane, or the sine of its distance from the great circle whose position we take. The value of p is p = ax + 3y + cz and the square of its distance p* = a^-^ + 3^y' + c^z^ + aa^xY + 2acxz + 25cyz If we square the p's for all the stars, and, using the notation of least squares, put \aa\ = 2a* [a3] = 2a5 etc. etc. P=12p^ we shall have 2P = [aa] x'' + 2 lai] xy + [db'] y'' + 2 [ac] xz + 2 [ 3c] yz + [cc] z= (i) The condition that P shall be a minimum, or dP= o, gives dP= iiaajx + [a3]Y + [ac]z)dx + ([a3]x + [33]y + \bc\z)dn + ([ac]x + [3c]y + \cc\z)dz = o (2) X, Y and z being subject to the condition X2 + Y^ + Z'' = I and therefore their differentials to the condition sidx + Y^Y + zdz = o (3) (5) 4 PRINCIPAL PLANES OF THE STARS. We now treat the equations (2) and (3) by the Lagrangian process of multi- plying (3) by an indeterminate coefficient X, subtracting the product from (2), and equating the respective coefficients of dx, dr and dz to zero. We thus have the three equations ( [a«] — \) X + [ai] Y + [ac] z = [ad'jx + {[dd] — X)y + [ic'jz = o (4) [ac]x + [^c]y + ([cc] — X)z = o These three equations and (3) suffice to completely determine the four unknowns X, Y, z, X. To find the latter we remark that the elimination of x, y and z from (4) gives a determinant equated to zero, namely [aa] — \ [a3] [ac] [ad] [53] - \ [be] [ac] [dc] \cc\ — \ This is a cubic equation in X, the roots of which are all known to be positive and real, giving three sets of values of x, y and z. For one of these sets P will be a minimum, for one a maximum, and for the third, neither. For each there will be a great circle satisfying the condition P = o. § 3. Relations of the Three Planes Given by the Solution. It is of interest to investigate the relations of the three great circles thus defined. To do this let us put \ , X' ; any two of the roots, X. ; x , y , z and x', y', z' ; the two sets of values of x, y, z corresponding to these values of X. Also, for brevity, we shall write A = laa-] B=[bb-] €=[00] J? = [ab] £:=[ac] 7^= [ic] (6) The equations (4) become by these substitutions (^ — X)x + Z>y + ^z = o Z»x + (^-X)Y + i^z = o (7) ^X + i^Y+(C-X)z = The other root, X', gives the corresponding equations (^ - X' )x' + By' + Ez' = o Z>x' + (^ - X' ) y' + Fz' = o (7)' ^x' + /^y' + (C-X')z' = o PROPERTIES OF THE PLANES. 5 We consider the relative positions of the two planes as defined by the cosine of the angle between them. What we want is therefore the value of the quantity cos / = xx' + yy' + zz' I being the required angle. To find this angle, multiply the first equation of (7) by x' and the first of (7)' by X and subtract. We have thus (V -X)xx' + Z>(x'y - xy') + ^(x'z - xz') = o Next take the second equation of each set and proceed in the same way. Multi- plying the second of (7) by y' and the second of (7)' by y' and subtracting we have Z)(xy' - x'y) + (\ - \')yy' + Fi^'z — xz') = o If we take the last equation of each set, multiplying the third of (7) by z' and the third of (7)' by z and subtract, we shall have a third equation of similar form. If we add the three equations thus formed, the coefficients of D, E and F will all be found to cancel each other and we shall have left the equation (X' — \)(xx' + yy' + zz') = o or cos T=o (8) Hence, the three ■planes defined by the equations (4) are at right angles to each other. We shall call these planes \}ci& -principal planes of the system of stars by which they are determined. § 4. Properties of the principal Planes. We have now to develop some remarkable theorems connected with these planes. First let us suppose that instead of referring the positions of the stars to an equatorial system of co-ordinates, we take the principal planes themselves as co-ordinate planes. Let us put «', 3', c', the co-ordinates of a star referred to the axes of the principal planes. The values of these quantities are then given by the system of equations «/ = 0X + ^Y + cz 3' = «x' 4- 3y' 4- cz' c' = «x" 4- ^y" 4- cz" (9) the accents on x, y and z distinguishing the three sets of values of these quantities. PRINCIPAL PLANES OF THE STARS. Forming the squares and products of a' and b' as we have already formed those of a and 3, and summing for all the stars, we shall have, by using the notation (6) \a'a'^ = [^]x^+ [B']y^+ [C]z^+2[Z)]XY+2[^]xZ + 2[i^]YZ [a'd' }={Ax + Z>Y + Ez) x' + (Z>x + ^y + ^z) y' + {JSx + J^r + Cz) z' But the equations (7) give Ax + Dy -\- Ez = \x Z>x + By + Ez = Xy Ex + Fy + Cz = Xz so that we have la'd' ] = X(xx' + yy' + zz' ) = o (10) In this way it is shown that, when we use the principal planes as those of refer- ence, all the terms of the determinant (5) except the diagonal ones vanish, and the equation for X becomes ([aV]-X)([3'^']-X)([cV]-\) = o of which the roots are \= [aV] X'=[d'd'-\ X"=lc'c'-\ (11) § 5. The Roots of the Fundamental Cubic. It does not follow from anything we have yet shown that the values (11) of X de- rived from the transformed system have any simple relation with the original values derived from the equatorial system. We shall now prove the remarkable theorem that the roots X , which are functions of the co-ordinates a, d ,c are invariant under any orthogonal transformation of these co-ordinates. To do this we may start from any system we please. Let us then start from that in which the co-ordinates are referred to the principal planes. Taking any other system A , let the coefficients of transformation, which are the cosines of the angles that the axes of the system A make with the principal planes, be the following: (12) We then have, for the values of a, d, c, for any star referred to the system A a=aa' + ^V + 7c' b = a!a> + ^'b> 4- 7'c' (13) c = ai'a' + ^"b' +y"c' a /3 7 a' /8' 7' a" y8" 7" THE ROOTS OF THE FUNDAMENTAL CUBIC. 7 From (13) we now form the values of [aa], [ad], etc., by taking the sum of the respective squares and products. We have a^ = a" a'' + fi'b'^ + y^c'' + 2a^a'b' + etc. ab = aa.'a'' + /3/8'3'' + (a^' + a>^)a'b' + etc. Summing with respect to all the stars and noting that, as already shown, \a'b']=o [«V] = o [3V] = o we have [aa] = a' \a>a' ] + ^ \b'b> ] + 7^ [c'c' ] , [ab] = aa' [a' a' ] + ^/3' [b'b' ] + 77' [c'c' ] etc. Using the abbreviated notation (6) the complete values of the six combinations are found to be A = aA' + 0'B' + rC' B = a'^A' + fi'^B' + y'^C C = a"' A' + ^"'B' + 7"'C' D = aa'A' + ^^'B' + 77' C E= aa"A' +^0"B' + 77" C J^= a' a" A' + ^'fi"B' + 7'7"C' where A' = [a' a'] etc., as in (6). What we have now to investigate is the relation which the roots X, X' and X" of the equation ;; (14) D B — \ F E F C—\ = o (IS) bear to A', B' , C, when A, B, C, etc., are defined by the equations (14). We have to express the roots of this equation in terms of A', B', C. This we do by finding the coefficients of the several powers of X in the equation (15). The coefficient of X* is evidently — i. To find that of X^ we remark that since no product containing n— 1 diagonal terms and no more can enter into a determinant of the nih. degree, the coefficient of X^ in the determinant is that which enters into the product of the diagonal terms. This coefficient is therefore A + B+ C To deduce its value we note that since a' + b'^c^^i 8 PRINCIPAL PLANES OF THE STARS. we shall have A + B + C= \_aa] + \bb'\ + [cc] = n (i6) n being the number of stars under consideration. Hence the coefficient of )^ is n simply. The coefficient of X. is found by developing the determinant to be D-" + E^ + F^ - AB - AC - BC Substituting the values of A, B, etc., from (14) and noting the orthogonal relations of the form {a^' - a'/8) = 7== we find the coefficient of \ to be - A'B' — A' C — B' C. Finally the absolute term of the equation is the determinant A D E \= D B F E F C From the values (14) we see that this Aq is the product « /9 7 aA' ^B' 7C' a' ^' 7' X a'A' ^'B' iC a" /S" 7" a"A' ^"B' i'C of which the second factor is equal to a /S 7 a' ^' 7' X A'B' C a" ^' ' ') f" Thus the term sought \?, ABC into the square of the determinant of transforma- tion (12), which, by a fundamental property of the determinant of an orthogonal system is equal to i. Thus the equation (15) takes the simple form {A! -\){B' -\){C' -\) = o (17) from which the coefficients of transformation (12) have entirely disappeared. It follows that the roots \, X', X." are invariant under this transformation, being always A' , B', C. We have therefore the remarkable theorem: The roots of the cubic which determine the -principal -planes of a system oj stars depend only on the arrangement of the stars on the celestial sphere, and are independent of the axes to which the positions of the stars are referred. INVARIANCE OF THE ROOTS. § 6. Professor Moore's Generalized Demonstration of the Invariance of the Roots of the Cubic Under an Orthogonal Transformation. ' The foregoing theorem is an algebraic one belonging to the theory of invariants. I therefore submitted the question of an algebraic demonstration to Professor E. H. Moore, Chicago, in the following form : Given, a triple system of quantities «i > bi, c . , ■ • • (« = 1, 2, •••,«) which are, in fact, the cosines of the angles which n lines make with the axes (rectangular) ot JC, Y and Z , and so satisfy the relation By the least-square process we form the quantities : [««] = 2a^ [a3] = 2a3 [ac] = 2ac ; etc. and then form the cubic equation in X [fliffl] — \ [a;3] [ac] A = {ab'] [bb'] - \ \bc\ \ac\ \bc\ \cc\ — X which will have three roots. Now, let us make an orthogonal transformation on the original a,b ,c,hy refer- ring to new axes. That is, we use, instead oi a, b , c a' = aa + ^b -{■ 'ic b' = a' a + ^'b + ic c' = d'a + ^>'b + i'c (a, yS, y) being the direction-cosines of the new axes. With these a', b' , c', we form \ad\^ [ab]', etc., by the same process as before, and then the determinant equation [aay-X' [ab]' [ac]' A'= [ab]' [bb]'-X' [be]' =o [ac]' [be]' [cc]'-X' to show that \' = \ that is, the determinant A is an invariant under the transformation. lO PRINCIPAL PLANES OF THE STARS. Professor Moore's demonstration is the following: "Write for a.b.c. («=I, 2, •••, ») Xj J Xj , Xj " Introduce " Consider the quadratic form $r(x^x,X3) = E ^/ = (say) E S'., ^x,X;, "Then (?/£>. = ?Ak i "1^ = 1.2.3) " Now the cubic \aa\ \ab'\ [«.] ^U S'W ?'l3 \_ah-\ \bh-\ [he-] are $'21 5^22 ^23 \ac-\ \bc] \.cc} ^31 ^32 ?33 ^11- ■ \ 5^12 ^13 ^2 I 6^. .--^ $^23 ^3 I ^32 5^33-^ " is the discriminant-invariant of weight 2 of the quadratic form ^(x^X^Xj) = ^(XjX^Xj) - \(x,' + x/ + Xg^) and so under any transformation of determinant ± i of X1X2X3 in both forms (X1X2X3) and x/ + X2^ + Xg^ it is unchanged, and further under orthogonal transformation (which is necessarily of determinant ± i) of ^(xiX2X3) alone it is unchanged (since Xi^ + X2^ + X3^ is thereby unchanged). " And this last remark covers the question, since a general orthogonal transfor- mation on the X1X2X3 induces on the coefficients ai^a'ae^si of each linear form «iiXi + «2iX2 + «3iX3 the corresponding contragredient (likewise a general orthog- onal) transformation. "Jn this phrasing the X1X2X3 enter as auxiliaries to effect convenient connection of the 0j 3j Cj a d c n n n which are initially not closely connected analytically, although geometrically they are in the relations covered by the theorem." properties of the principal planes. ii § 7. Properties of the Principal Planes. Our next step is to distinguish between the properties of the three principal planes, with respect to the corresponding values of P' . By multiplying (3) by X and subtracting it from (2) we have a certain value of dP which may be written in the form dP = Ld^ + Md^ + Ndz (18) where Z, J/ and N have the values given below. The quantity on which our conclusions must now depend is the second derivative of /*, or, d^P= Ld^x + Md^Y + Md^z + dLdx + dMdY + dNdz The principal planes themselves have been determined by the conditions Z = o M= o Ii= o Hence, for those special values of P corresponding to the three planes we have rf=i'= rfZ^x + dMdY + dNdz The values oi L, M and N are Z = (^ - \)x + 2?Y + Ez M= Z>x + (^ - X)y + Fz ir= Zx 4- i^Y + (C - \) z Differentiating these values and substituting the result in the preceding equation we shall have d^P= (4 — \)dx^ + {B — X)dY^ + {C — X)dz^ + 2DdxdY + 2Edxdz + iFdYdz This equation is valid for any rectangular system of co-ordinates. Its investi- gation is simplified when we suppose the system to be that of the principal planes. We then have from (10) Z> = o Z=o Z'=o (19) and the last equation becomes d-'P = {A! - \) dx? + {B' -\)dY^ + {C' -\) dz\ (20) Let us take for the roots \ X = A' \' = B' \"=C' (21) The three corresponding values of d'^P axe then (a) ; d-'P^iV - \) dY" + ( \"-X) dz^ {b) ; d^P={\-\') dx? + {\" - X) dz" (22) {c); d^P={-k-\')dy? + {\' -\")dY^ 12 PRINCIPAL PLANES OF THE STARS. Let US suppose the values X, X', X" to be arranged in the order of magnitude, so that \x>o) In this case c^ will be contained between the limits X** < «^ < x" + 2X(/X The mean value of a^ will then be f xV^= r ; 1/0 I/O as we already have seen from geometric considerations. 14 PRINCIPAL PLANES OF THE STARS. The deviation of any actual value, x^ , from this mean value is and its square is i-K + x^ The mean value of this square is The mean sum of the squares of all the deviations for the n values of c^ is e^ = 4« 45 the square root of which is the probable mean deviation of any one X. Hence we may write X= -±0.298 i/« (25) Unless the smallest X fall below the inferior limit thus indicated, we cannot regard the evidence of a tendency toward any plane whatever as strong. The farther it falls below, the stronger it becomes. § 9. Solutions of the Equations. We have now to return to the solution of the equations (5) which give the values X, y, z, on which depend the position of the galactic pole, or of the pole of the principal plane of condensation. It will be convenient to recapitulate the numerical operations ab initio. For each of the n stars of the given system we compute a = cos S cos a b= cos S sin a c = sin S Here 3-place logarithms are ample and 2-place might do well. Then compute As a check we have A-\- B ■\- C= n. We next form a cubic equation in X, of which two of the coefficients are: p==AB + BC+ CA-D^-E^-F^ g = ABC + iDEF- AF' - BE' - CD' The complete equation is then \^—n\'+;p\—q=o (26) n being the number of stars. Only the smallest root of this equation is required to determine the plane of condensation, but it will be of interest to have the other two roots. SOLUTIONS OF THE EQUATIONS. 1 5 Having solved this equation we put A" = A-X B" = B-X C"= C-X and shall have, to determine x, y, and z, any two of the equations A"x + Z>Y + ^z = o Z>x + B"y + Fz=^o (27) ^x + i^Y + C"z = o and the equation X^ + Y* + Z^ = I It is not, however, necessary to have the actual values of x, y and z or even to write out the preceding equations. If we put Ai, D^, the Right Ascension and Declination of the galactic pole, we have cos Z>j cos ylj = X cos Z)j sin A^ = Y sin D^ = z . Y - z sin A, z cos A, , „. tan ^1 = - tan D^ = * = — * (28) whence Owing to the vanishing of the determinant of (27), these quantities may be expressed in various forms as functions of the coefficients. It is advisable to choose that form which depends on the largest numbers; or we may use and compare the results of various forms. We find in fact from the solution EF-C"D E^- A" C" DE- A"F tan ^1 - ^„ ^„ _ ^2 _ ^„j^ _ ^p - j^p _ ^„p (29) If the process is rigorously correct these three values should be equal. But the errors arising from omitted decimals will necessarily produce a discrepancy. We may compare the three difterent values to detect any large error and should prefer that fraction having the largest terms. We have also tan Z>i _ Z)' - A"B" _ B"E-DF _ DE-A"F E^T^l ^ B"E -DF~ F^- B" C" ~ EF- C"D (30) sin^^ DE-A"F C"D-EF A"C"-E^ ^^ ' tan Z>i ~ A"B" - D'~ B"E - DF ~ ED - A"F among which we may choose the best for determining D^. The values of Ai and A thus found will be the Right Ascension and Declination of the pole of the principal plane, or plane of condensation of the system. i6 PRINCIPAL PLANES OF THE STARS. § lo. Position of the Central Plane of the Galaxy. We now proceed to the applications of the preceding method and begin with a determination which is the fundamental one — that of the position of the galactic plane itself. This I have done by marking a series of points throughout the length of the galaxy, conforming as closely as practical to the following conditions : 1. The points to be located either on the central line of the galaxy, as fixed by eye-estimate, or in the centers of the agglomerations. 2. The distance apart of consecutive points to be about io° or less. 3. The positions of the points to be completely independent of each other; that is, no bias to be allowed which would tend to bring any one point into line with the others. In applying the first of these conditions a difficulty is encountered in the great bifurcation between Cygnus and Aquila. Here two streams may be followed, of which the preceding one terminates south of the equator after a wide divergence from what seems to be the main stream. It is a question whether this divergent branch should be included in the determination. This question can best be grap- pled with if we make two determinations, in one of which the branch is omitted, and in the other included. In laying down the points I used Heis's Atlas Ccelestis for the northern hemi- sphere, and Gould's Uranometria Argentina for the southern, supplementing the first by naked-eye observations made from time to time through several seasons. The system of points finally settled upon was the following: Main Stream of the Galaxy. Branch. R. A. Dec. R. A. Dec. R. A. Dec. R. A. Dec. 0° + 65° 117° -25° 267° -32° 272° + 4° 10 62 121 -30 268 -35 279 10 25 60.5 123 -38 270 -28 284 20 40 57 126 -47 275 — 20 288 28 58 50 135 -53 277 - 15 298 + 35 65 42 150 -58 279 - 7 75 37 165 -61 287 83 28 180 -63 293 4- 11 85 25 200 -63 298 20 87 18 212 -61 306 30 97 + 8 225 -55 306 40 103 - 5 232 -50 314 45 no - 9 250 -43 334 54 "3 — 20 255 -32 , 250 + 62 These positions give the following two sets of values of ^, B, etc. The first set is reached when the branch points are omitted, the second when they are included. ■?i55:^ POSITION OP THE GALACTIC PLANE. 1 7 First Set. Second Set. First Set. Second Set, A 5.244 5-554 D — 3.070 - 3-969 B 19.568 23-569 B + 8.448 + 8.904 C 17.190 17.892 ^ +2-535 + I.I74 Sum 42.002 47.005 The respective equations in X, with the smallest root in each case, are (i) X* — 42X^ + 441. 92\ — 40.24 = o; \ = 0.092 (2) X'— 47\* + 559. i4\ — 96.82 = 0; \ = 0.176 The positions of the galactic pole hence derived are (i) ^i=i92°.8; 2?.= + 27".2 (2) ^, = 191 .1; 2?i = + 26 .8 ^ There is therefore a difference of somewhat more than one degree in the position of the pole according as we include or omit the branch. The results will now enable us to consider the question : Is the central line of the galaxy sensibly a great circle P To do this we form the galactic latitude of each of the points which we have marked down as determining points of the central line. If the central line is a great circle, the mean galactic latitudes of all these points should reduce to zero. The galactic latitude, ^, of a point (a, 6, c) is given by the equation : sin /3 = ax 4- 3y + cz where the six quantities have the values assigned. Using the first position of the galaxy, which will suffice for our present purpose, we have logx = 9.938„ logY = 9.292„ log z = 9.660 Thus the expression for sin /8 may be written — sin /8 = [9.938] cos a cos S + [9.292] sin a cos S — [9.660] sin S This gives the following values of the galactic latitudes of the 42 points which we have marked down as determining the galaxy. In column Br. we give the latitudes of the branch-points, also measured from the first position of the Galactic circle, in the determination of which they are not used. /8 P P P fi P Br. + 2°.7 -2°.9 + o°.3 -o°.7 — 2". I -4°.o + ii"-3 -0 .7 -3 .0 + .7 -0 .4 -4 -I -3 -7 + 7-8 -I .7 -5 .0 — 2 .2 + .4 -2 .4 -4-3 -f 8 .0 -2 .7 -I .4 -5-3 — I .2 -2 .5 + 1 -5 + 8 .2 -3 -o — 2.1 -4.6 + 1 .2 -I .8 — .2 + 4 .1 -5-7 + 2 -3 -2 .4 -2 .4 + .1 — 2.1 -3 -2 -0 .4 — I.I -0 -5 -3 .8 + 1 .1 1 8 PRINCIPAL PLANES OF THE STARS. Omitting the branch, there is an obvious preponderance of negative signs, only 9 points being in north latitude, against 33 in south latitude. The mean value of the latitude of the 42 points of the main stream is — i°.74. It therefore appears that this stream is not, on the whole, a great circle, but has a deviation showing a small, but well-marked displacement of our system from the central plane in the direction of the constellation Coma Berenicis, in which the north galactic pole is situate. We have next to inquire how far this result will be modified if we regard the branch as entitled to equal weight with the mean stream. In this case the solution we have found changes the position of the pole about i°.5 of a great circle. But the exceptional deviation of the branch, amounting in the case of one point to 10°, still subsists. Notwithstanding this, I conceive that for cosmological purposes the branch should be included, which will lead us to the second position of the pole- To determine the mean deviation from a great circle when the branch is included, it is not necessary to compute the individual values of /8 for the second circle, since this mean will not be altered except from unequal distribution of the selected points around the circle. The inequality will be very slight if we compare the five branch points with the five points nearest them on the main stream, taking the mid-points of each pair as new ones to replace both of the others. By this process we find Mean/8= — o°.98 An appreciable deviation from a great circle therefore still remains. § II. Gould's Belt. Sir John Herschel * first called attention to the fact that while a number of the brightest stars in the heavens lie near the course of the galaxy, they systematically deviate from it in the direction of a great circle cutting the galaxy at two nodes, in Cassiopeia and Crux respectively, and making an angle of nearly 20° with it. This relation of the stars in question was more fully investigated by Gould f who by a process based on the judgments of himself and his assistants fixed the pole of their central circle in the position R. A. = ii''2S"' Dec. = + 30° This medial circle was estimated to be a small one less than 3° from the parallel great circle, and on its southern side. It was termed by Gould " the Belt^'' in order to distinguish it from the galactic or central circle of the Milky Way. Gould also investigated the distribution of the bright stars to mag. 4.0 with respect to the Belt and the galactic circle respectively. Finding for each star its * Expedition to the Cape of Good Hope, p. 385. t Uranometria Argentina, p. 355. Gould's belt. 19 angular distance in degrees from each circle, he took the sum of the squares of each set of distances with the result: Number of stars 527 Sum of distances from Belt 14210° " " galactic circle 14972 squares of distances from Belt 627578 " " " galactic circle 653602 Gould hence draws the conclusion that the stars in question may form a system by themselves, to which our sun belongs, and of which the principal plane tends towards the Belt rather than toward the galactic circle. We can better judge the significance of the result if we determine the probable sum of the squares of the distances of 527 stars scattered at random from any circle taken at pleasure. The distribution being supposed uniform, the area of a girdle of breadth ds at a distance 8 from the great circle will be da = 27r cos S dh The number ol stars contained in this area may be expressed in the form dn = kda = 2irk cos SrfS k being the star thickness per unit of area. The sum of the distances of these stars from the great circle will be hdn = iirkh cos S dh and the sum of the squares ^dn = 2nkS^ cos SdS The integrals of these quantities for an entire hemisphere will be : For total number, JV, of stars cos BdS= /^.irk (a) For the sum of their distances : 8 cos BdS= 2.28327r>fe (b) For the sum of the squares of the distances : 8^ cos S rfS = (tt" - 8)7r^ = i.Seg&irk (c) We use the first equation to replace k by iV in the next two equations, putting 47r To make our results comparable with those of Gould, who used the degree as the unit, we must multiply (^) by 57°. 296, and (c) by the squares of this number. 20 PRINCIPAL, PLANES OF THE STARS. We thus find that the probable value of the sum of the distances of JV stars, dis- tributed at random over the sphere, from any great circle is 2=[i.si46]°i\^ and the sum of the squares of the distances is 2.= [3-1859]°^ For JV= 527 we have 2 = 17230° 2j = 808500° Both numbers are enough greater than the actual ones found by Gould to show a marked tendency of the stars in question to crowd towards the Belt, or the galac- tic region. But it does not seem certain that the slightly greater tendency toward the Belt than toward the galaxy is the result of a general cosmological law. The writer conceives that we should rather regard it as the result of a remarkable col- lection of stars of extraordinary luminosity in the region of Orion and the constel- lations south of it, near the galactic region, but not included in it. All the brighter stars in this region, Sirius excepted, are notable for the minuteness of their proper motion, generally from i" to 2" or 3" per century. The great distance thus inferred is strengthened by the failure of Gill and Elkin to find any measurable parallax for Canopus, Rigel or ^8 Orionis. The simple fact seems to be that we here have an exceptional number of stars of extraordinary absolute luminosity, probably tens or hundreds of thousands of times that of the sun. We shall next apply the methods of the present paper to the determination of the principal planes of the brighter stars. § 12. Principal Planes of Stars of Various Classes. A. Gould's Belt. I have taken 48 stars of magnitude brighter than 2.5 as determining stars of this belt. From these are taken out 12 having a proper motion so large that we cannot regard their present positions as of cosmological significance, leaving 36 to be used. The following are the resultant numbers which determine the principal plane of the stars retained: yl = 4.683 ^=16.294 (7=15.023 Z?= — 0.396 ^= + 6.796 7^= — 1.024 \^ — 36\'' + 344X — 396 = o X= 1.329 For pole of principal plane A=i79°.6 Z>i = + 26.4 PRINCIPAL PLANES OF STARS OF VARIOUS CLASSES. 21 This is 11° distant from the position we have found for the pole of the galaxy. The inclination of the belt to the galactic plane has therefore little more than half the value found by Gould. B. Two other poles were determined, the one for all the stars to mag. 2.5, the other for all to mag. 3.5, none being excluded in either case on account of proper motion. The results were 78 stars to mag. 2.5 ; ^=181°. 2 D=j.*j°./^ All " " «' 3.5 180 .0 21 .5 C. A determination was also made for all the lucid stars. Owing to the great labor of using all the stars individually as the basis of the determination, I took as fundamental data the star richness given on Schiaparelli's planispheres. Here the whole sky is divided into 36 zones of declination, each 5° wide, and these again into trapezia, by suitable circles of right ascension, of a general average length not differing greatly from 5°. In each of these trapezia is marked the mean star-rich- ness, per 100 square degrees, of the trapezium itself and the adjoining ones. I used these numbers in the following way. Let us put U, the area of a trapezium; a, 8, the Right Ascension and Declination of its center of gravity; hy its mean star density; n^ hU, the number of stars it is taken to include. We then proceed on the assumption that there are n stars in the position (a, 8). Forming the values of a^ , ab , b"^ , etc., for an entire zone of any declination and summing, we shall have [aa] = cos^ S2« cos^ a [«c] = sin S cos S2« cos a \ab^ = cos^ S2m sin a cos a [^c] = sin S cos S2« sin a \bb'] = cos^ S2« sin* a \cc\ = sin* S2m Let n' be the value oi hU for the corresponding zone in the southern hemi- sphere. The numbers for the southern zone will then be found by substituting n' for n , and changing the sign of 8 . Doing this and adding the results to those for the northern zones, we find that two zones, north and south, of Declinations + 8 and — 8 contribute the following values to the coefficients : \ad\ = cos* 82 (« + «') cos* a [ac] = sin 8 cos 8 (« — «' ) cos a \ab'\ = cos* 82 (« + «') sin a cos a [^c] = sin 8 cos B(n~n') sin a [3(5] = cos* 82 (« + «') sin* a [cc] = sin* 8 (« + «' ) The complete values of the coefficients are then found by summing with respect to all the values of 8 taken positively. 22 PRINCIPAL PLANES OF THE STARS. In forming the numbers, zones io° wide were used, and in each zone the trape- zia were twice as long as those of Schiaparelli, that is, io° near the equator. It may also be remarked that the values of 8 were taken for the mid-points between the boundaries of the trapezium instead of for the center of gravity. The results of the summation are: [aa] = 867 [ac'] = + 101.9 [ab} = — 6.0 [6c'] = — 14.3 [33] = 1053 [cc} = 1078 \= 825.9 V = 1049.3 \"=II22.4 The position of the pole is A^ = i8o°.o Z>i= + 2i°.s This pole is nearly 10° from that of the galaxy, and falls so near the one which we have found from the brighter lucid stars as to suggest a connection among the lucid stars in general, and a systematic deviation of their principal plane from that of the galaxy. But before we assign any significance to this deviation, another fact of cosmological importance is to be considered. D. The deviation of the principal plane of the lucid stars from that of the galaxy as just found suggests the question whether the same may not be true of all the non-galactic stars. Were this true, it would show a remarkable difference in the arrangement of the galactic and non-galactic stars. To decide this question it is not necessary to go through the great task of determining the principal plane of all the stars to the ninth magnitude. It will suffice to determine the richness of the sky in these stars in regions where the separation between the galaxy and the " Belt" or other principal plane of the bright stars is greatest: that is, the region of Orion and Canis Major on the one side, and that between Aquila and Cygnus on the other. On each side of each of these I took a trapezium 10° broad and 15° of R. A. in length, and counted the stars of the Durchmusterung within its limits. Owing to the remarkable richness of Orion in bright stars, I took another trapezium 8° broad and 15° in length including most of that constellation. The results are tabulated thus : — :ral Point of Region. Galactic Latitude. Richness. Distance from Belt. 5" 30°" - 6° -19° 20.7 6° 550 -15 -19 22.8 6 830 - 6 + 20 23.8 35 18 10 + 30 + 20 18.2 33 21 10 + 20 — 20 19.2 4 PRINCIPAL PLANES OF STARS OF VARIOUS CLASSES. 23 It will be seen that the richness is equal at equal distances on the two sides of the galactic circle, and has no relation to the belt or to the equator of condensation of the bright stars. To all appearance the principal plane of all the D. M. stars coin- cides with that of the galaxy, or, at least, does not deviate in the direction of the Belt. It would seem from this that the deviation of the principal plane of all the lucid stars from that of the galaxy grows out of the features already mentioned in the case of Gould's Belt. E. The Fifth-type Stars. The remarkable fact that the so-called Wolf-Rayet or fifth-type stars are mostly very near the central line of the galaxy was brought out by Pickering. The only exceptions are that a few of these stars are tound in the Magellanic Clouds. Professor Pickering kindly communicated all the known positions of such stars, not in the Magellanic clouds, 7 1 in number. From these positions were derived: ^ = 10.051 ^=25.040 (7 = 35.906 Z)=— 3.709 ^= + 17.842 /^= + 2.098 \^ — 7iX^ + 1175. 3X — 250.6 = o \= 0.216 V = 25.770 V = 45.014 A^= i90°54' Z>i = + 26°39' This position deviates by only 15' from that which we have found for the galactic belt. It would be interesting to decide whether the positions of these stars seem to be fixed with respect to the galactic circle simply, irrespective of the galactic agglom- erations, or whether they belong primarily to the latter, and lie near the galactic circle only because the agglomerations to which they belong lie near it. I have noted the brightness of the galactic region in which each star is situated, but the result is indecisive. While a majority of the stars are in the brighter agglomera- tions, there are a score in the fainter regions. The mean galactic latitude of the stars is as great as that of the brighter agglomerations. The question is still farther complicated by the unequal distribution of the stars in question, more than two- thirds of which are in south declination. The positions of the various poles we have determined are R. A. Dec. Galactic plane (omitting branch) igz^.S + 27°.2 " " (including branch) 191 .1 26 .8 Gould's Belt, as found by Gould 171 .2 30 The Belt as determined from 36 bright stars of small p.m. near it 179 .6 26 .4 Plane of all stars to mag. 2.5 181 .2 17 .4 " " " " " " 3.5 180 .0 21 .5 " " fifth type stars 190 .9 26 .7 24 principal planes of the stars. § 13. Law of Richness of the Galactic Region. We now continue our study in another detail. Our formulae determine the principal plane of a system of stars, but do not decide as to the law of variation ot star-richness from the pole to the great circle of the plane. Especially they do not decide between a mere belt of stars around a sphere in which the stars are equally scattered everywhere outside the belt, and a continuous law of increase from the pole. In the case of the fainter stars it is well known that the star-thickness is least around the galactic poles, and increases, slowly at first, then more and more rapidly, as we approach the galactic belt. The law of increase is, so far as we can judge, that which we should find if the universe were a comparatively thin stratum of stars, of which the boundary is near enough to our supposed central position to admit of our seeing at least the brighter of the stars at the boundary. Considering only the distribution of the stars in galactic latitude, we might infer that they are equally thick throughout all space, and that their greater apparent thickness in the galactic region is due wholly or mainly to the fact that we here see through a greater depth of the stratum. But a study of the structure of the Milky Way shows that this is not the whole truth. The many rifts and clusters of stars in this region show that, besides a possible uniformly distributed universe of stars, we have, surrounding us like a girdle, a great number of irregular agglomerations of stars, of very varied thick- ness and extent, having in some cases a fairly well-marked boundary. It seems probable that the stars of these agglomerations are distinguished from other stars by their blue spectral type and the wide range of their absolute lumi- nosity. We must therefore in any study of this subject divide the stars into two groups — those which form the agglomerations of the Milky Way, and those which pertain to the universe at large. It is not as yet possible to make this division with sharpness in the case of indi- vidual stars. Statistically, however, an approximation may be made. Easton * has shown, by comparing the star-richness in the galactic agglomerations, that the brighter the latter the more stars to the ninth magnitude they contain. The writer f has shown that this is true even of the lucid stars. This was done by comparing the star-richness per square degree in the darker galactic regions with that in the agglomerations. The remarkable fact was thus brought out that, in the darker galactic regions, the richness in stars to the seventh magnitude is no greater than it is in the non-galactic sky. This fact suggests the question whether there is any general increase of richness of lucid stars as we approach the Milky Way. We may examine this question by Schiaparelli's planispheres. On these were taken * Astrophysical Journal, I, p. 216. f The Stars; a study of the Universe, New York, 1901. LAW OF RICHNESS OF THE GALACTIC REGION. 25 four regions, one of 45 ° radius around each galactic pole, and two zones, each con- tained between 20° and 45° of galactic latitude. From the southernmost of these zones was excluded the dense collections of stars extending from Taurus to Orion. The results for richness per 100° were: North. South. Star richness round galactic poles 7.83 8.36 " " of mid-zone 8.82 9.50 This slight increase may be partly attributed to the extension of galactic stars in certain regions beyond the limit of 20° of galactic latitude. The systematic increase, while probably real, seems to be slight. The conclusion is that if the galactic agglomerations were excluded from consideration, the crowding of the lucid stars toward their principal plane would be scarcely, if at all, greater than we might expect to find as the result of the irregularity of chance distribution. It is also to be remarked that, in such a case, we should probably find an even greater inclination of the principal plane to that of the galaxy than we do in the case of Gould's Belt. The question next arises whether we can form any idea of the magnitude of the increase of richness from the galactic poles to the Milky Way in the case of the fainter stars, say those of the Durchmusterung. This increase may be termed an ellipticity of the stellar system. Considering the stars in the galactic region as of the two classes already mentioned, the ellipticity of which we are in search is only that of the non-galactic stars. As we cannot at present completely separate the latter from the others, we must be satisfied with a determination of the star- richness in the darkest regions of the galactic belt. I have continued Easton's work, already cited, by finding the richness given by the Bonn D. M. along the darker portions of the great cleft from Cygnus to Sagittarius, and that given by the Cordoba D. M. in some southern clefts. The regions where clefts were indicated were selected from the atlases of Heis and Gould. In the following table the first column gives the zone, 1° in breadth; the second the R. A. of the beginning of the strip; the third the length of the strip in R. A.; the fourth its area, and the fifth the number of D. M. stars which it contains. In the regions G and H this number is that catalogued in the Cordoba D. M. In K it is that of the Cape Photographic D. M. The latter region is that known as the " coal sack," In G and -ff"we have taken smaller regions, G' and K' yet poorer in stars, so that these regions are duplicated in G and K. But in K' the numbers are those of the Cordoba D. M., not the Cape, the stars in it extending to a markedly fainter limit of magnitude. 26 PRINCIPAL PLANES OF THE STARS. Table of Star-richness in the Darkest Regions of the Milky Way. leg. Zone. Beg. Length. Area. Star A 49° 21" 0"" 20" 3°-25 61 50 20 50 20 3.18 63 51 2045 20 3 -12 47 52 20 40 20 3 -04 74 53 2035 20 2 -97 80 54 20 30 20 2 .90 64 55 20 30 10 I .41 19 .87 33 422 B 51 3 10 10 I .56 24 52 3 10 50 7 .61 98 53 3 10 50 7 -44 95 54 3 10 50 7 .26 76 55 . 3 10 50 7 .08 84 56 3 15 45 6 .90 77 57 3 20 40 5 -38 43 -23 52 506 C 57 22 30 4 -04 115 58 22 40 5 -22 95 59 22 5 55 6 .99 166 60 22 10 70 8 .6i 173 61 22 20 50 5 -96 30 .82 119 668 D 31 20 20 4 .26 143 30 1955 25 5 -39 176 29 1950 25 5 -44 189 28 1945 25 5 -5° 20 .59 184 692 E 26 1940 10 2 .24 68 25 1935 10 2 .26 74 24 1930 10 2 .28 73 23 19 20 20 4 -59 "3 22 19 15 25 5 .78 132 21 19 10 30 6 .98 189 20 19 10 25 5 -86 29 -99 133 782 P 13° 19" 10" lO"" 2° -43 45 12 19 5 15 3 -66 65 II 19 5 15 3 -68 78 LAW OF RICHNESS OF THE GALACTIC REGION. 27 Reg. Zone. Beg. Length. Area. Stars. -F (continued) 10 10 20 4 -92 112 9 1855 20 4 -93 "5 8 1850 20 4 -94 117 7 18 40 25 6 .20 154 6 18 40 25 6 .21 "3 5 ,1835 25 6 .22 "3 4 1830 30 7 .48 169 3 18 25 30 7 -49 138 2 18 20 30 7 -50 142 I 18 10 30 7 -50 139 + 18 40 10 .00 165 — 1740 50 12 .50 160 — I 1730 60 14 .99 no .65 159 1984 a -25 17 20 20 4°-5i 130 -26 17 20 20 4 -48 209 -27 1725 15 3 -33 226 -28 1730 10 2 .20 231 -29 1735 5 I .09 15 .61 no 906 G' -24° I7''2l"' 6- i°.36 23 -25 17 21 6 I -34 9 -26 17 21 6 I -33 4_-o3 18 50 H -32 17 10 10 2 .11 196 -33 17 10 10 2 .08 153 -34 17 20 4 .12 8.31 205 554 K -60 12 50 10 I .23 86 -61 1235 25 2 .98 120 -62 12 25 35 4 -03 209 -63 12 25 30 3 -34 166 -64 12 25 25 2 .69 14 .27 133 714 K' -61 1234 20 2 .38 73 -62 1236 20 2 .31 85 -63 12 30 18 2 .00 79 -64 12 23 14 I .50 8 .19 44 281 28 PRINCIPAL PLANES OF THE STARS. The mean richness of the dark regions catalogued is as follows . Position of Position of Region. Central Part. Richness. Region. Central Part. Richness. R. A. Dec. R. A. Dec. A 20 50 + 52° 21.2 D.M. G 17 30 -27° 58.1 Cordoba D.M. B 3 30 + 54 II. 7 " & 17 24 -25 12.4 " C 22 30 + 59 21.7 " H 17 15 -33 66.7 " D 20 + 30 33-7 " K 12 40 -62 50.0 C.P.D. E 19 30 + 23 26.1 " K' 12 40 -63 34.3 Cordoba D.M. F 18 50 + 8 17.9 " Let us compare these numbers with those found by Seeliger from the Northern Durchmusterung.* From his table of mean richness in zones of galactic latitude 20° broad I interpolate the following table of richness for parallels of galactic lati- tude 20° apart. : Gal. Lat. Richness 80° 8.6 60 9.9 40 13.0 20 20.0 25.0 The mean density within a zone extending 10° on each side of the galactic circle is 24.6, while the mean for the first six dark regions above is 20. In the case of the Cordoba D. M. the mean richness around the South Galactic Pole is about 30. It is remarkable that with the exception of region B , in Perseus, and G , in Sagit- tarius, the star-richness is nearly the same in the darker regions as in the galactic region in general. The most remarkable feature of the case is that in the region G", covering 4 square degrees, there are only 50 stars of the Cordoba D.M., while on the theory of random distribution over the whole sky, ignoring any ellipticity, there should be more than 150. It is not necessary to undertake a computation of the probability that such a deviation as this could be accidental. The unavoidable conclusion is that we have, in this region, a vacant space, not only in the Milky Way, but in the universe of stars at large, independent of the Milky Way. Indeed, such vacant spaces seem to be the rule in all this region of the heavens. Returning now to the problem of what I have termed the ellipticity of star dis- tribution independent of the agglomerations of the Milky Way, we begin by inquiring what would be the result if we consider that those agglomerations do not *Mtinich Academy Sitzungsberichte, 1884, p. 544. THE PROBLEM OF TOTALITY OF STAR-LIGHT. 29 extend into the vacant spaces catalogued above.* The comparison of the richness of the sky per square degree in a large region around the galactic pole, and in the rifts of the Milky Way, as derived from counts of Argelander and Thome, will be : Bonn D.M. Cord. D.M. Neargalactic pole 8.6 29.9 In rifts of Galaxy 19.8 45.7 In the Galaxy generally 24.6 94.1 If we define the ellipticity by the ratio of star-thickness near the galactic equator, the agglomeratious being ignored, to that around the poles, it is clear that the results from the two catalogues will be markedly discordant. In the northern hemisphere the galactic density is more than double the other; in the southern hemisphere it is only fifty per cent, greater. The result must therefore be rather vague ; but, in the absence of any present data for a more exact estimate, the con- clusion may be stated as follows : Were the agglomerations of the Milky Way removed, -we should still find a continuous increase in the richness of the sky from the -poles to the galactic circle, -where it -would -probably be nearly t-wice as great as at the poles. It would seem therefore, that the ellipticity of the system is markedly less than would be inferred from the total number of stars in the galactic region. § 14. The Problem of Totality of Star-light, The preceding rather discordant results suggest that one of the important prob- lems of stellar statistics is the determination of the law of star-richness in various regions of the sky for stars of various classes, especially of the law of increase toward the galactic circle for stars of various classes. The end in view would be the deter- mination of the law of richness with such precision as to enable some estimate of the total light of all the stars to be determined. The fundamental determination resolves itself into that of the ratio of the number of stars of each magnitude m to the number of magnitude m— i. If we put r^^ for this ratio and ;' for the light ratio corresponding to a unit change of magnitude the totality of light received from all the telescopic stars within any region of the sky will be proportional to the sum of the series r'r^ + r'r^ • r'r^ + r'r^ • r'r^ ■ r'r^ 4- • • • * This investigation was made before the appearance of Easton's extension of his former work, based on comparisons of the density of the Argelander stars in various regions of the Milky Way defined by their brightness. As the numbers given by Easton are relative, they cannot directly be applied in the above problem ; I have therefore not attempted to utilize them at present. 30 PRINCIPAL PLANES OF THE STARS. SO long as t'r^> i or r„, > 2.512 the light from the totality of the stars of each suc- cessive order of magnitude will form an increasing series. The general result from counts of stars up to the present time is that this increasing ratio really subsists through several magnitudes of the telescopic stars, probably the 9th or loth; per- haps yet farther. . It must, however, speedily fall below the limit, else the total amount of light received from the sky would be greater than it is. The problem in view may therefore be defined as that of the determination of the series of ratios. to the highest attainable value of m. We know the value oi r' to be between 3 and 4 up to m = 6, and then to slowly diminish according to a law which is differ- ent in different parts of the sky, but has not yet been determined in any case. The material available for making this determination is voluminous, but not wholly satisfactory. There are also many investigations of this material in which the problems relating to the subject are considered from various points of view, but a work leading to a well-defined and specific conclusion is still wanting. The fol- lowing conspectus of the material and some of the investigations which are avail- able for the problem is presented with the hope that it will be found useful to any investigator able to attack the problem. No attempt has been made at complete- ness in the presentation; the works cited are merely those with which the writer has become acquainted, and which seem to have a direct bearing on the subject, or to contain results useful in a more extended work. 1. The gauges by the two Herschels, as found uncompleted in the Philosophical Transactions and in Sir John Herschel's Voyage to the Cape of Good Hope, and completed and reprinted by Holden.* One result of these gauges, as discussed by Seeliger,t is that, in the Milky Way, the richness of the sky in the faintest star visible in Herschel's telescope, presum- ably stars of about the 14th or 15 th magnitude, is twenty times as great in the galactic zone as around the galactic pole. 2. The gauges of Celoria, embracing the zone 0° to + 6 around the heavens.J These give a mean galactic density about 2.5 times as great as the density around the galactic poles. 3. The counts made by Pickering § of stars found on his photographs, and their statistical distribution according to magnitude. This work is still incomplete; but * Publications of the Washburn Observatory, Vol. II, p. 113. t Betrachtungen fiber die raumliche Vertheilung der Fixsterne, pp. 27-29; Abh. K. bay. Aka- demie der Wiss. II, CI. XIX. Bd. III. Abth. X Supra alcuni scandagli del Cielo; Pub. del Osser. d. Brera, XIII. § Harvard Annals, Vol. XL VIII, No. 5 (separate). THE PROBLEM OF TOTALITY OF STAR-LIGHT. 3 1 one conclusion is that the mean galactic density is little more than twice the polar density down to the nth or 12th magnitude. 4. The Bonn Durchmusterung, by Argelander and Schonfeld, extending from the north pole to 23° south declination. Several researches based on this work are to be cited. The investigations of Seeliger * lead to the conclusion that the ratio of progres- sion in the stars is markedly greater in the Milky Way than elsewhere, and increases in proportion to the density of the galactic agglomerations. The work also derives the star-richness in various zones of galactic latitude. Counts by StratanofF,t showing the richness of all portions of the sky in the Bonn and Cape Durchmusterung, will also facilitate the use of these works. 5. The Cordoba Durchmusterung,J now extending from —23° to —53° of decli- nation. The continuation of this work to its end at the south pole is a great dis- sideratum of astronomy at the present time. A needed improvement is the return in subsequent volumes to the scale of magnitudes adopted in the first two volumes, which seems to have been abandoned in the third. 6. The Cape Photographic Durchmusterung. It has seemed to the writer that the limiting magnitudes of this catalogue need revision. 7. The international chart of the heavens of which the zone +31° to +40° was undertaken by the Potsdam Observatory. This special zone is mentioned because three volumes of it are now complete.§ The limiting magnitude in this catalogue is supposed to be 11. o. Part I of the Potsdam publication shows the following degrees of richness per square degree : Near Galactic Pole JR = 16 Darker part of Galaxy (near 6h.) " 90 Dense agglomeration, near R. A., 19 h. 30 m " 222 8. Easton's investigation of the relative densities of the stars of the B. D. in the regions of the galaxy of different degrees of brightness. || 9. Various works dealing with the necessary problem of reducing the various systems of magnitudes in these catalogues to a uniform photometric standard. Seeliger has discussed the reduction of the zones of the Bonn Durchmusterung *1. c, p. 15 et seq. t Publications de I'Observatoire de Tachkemt, No. 2, 3, 1900. X Publications of the Cordoba Observatory, Vols. XVI, XVII, XVIII, § Publicationen der Astrophysikalischen Observatoriums zu Potsdam ; Photographische Himmels- karte, Band I, II, III. II La distribution de la lumiere galactique, compar^e k la distribution des 6toiles catalogu6es dans la voie lact6e bor^ale. Amsterdam Verhandelingen Eerste Sectie, Deel VIII, No. 3, 1903. 3? PRINCIPAL PLANES OF THE STARS. to the Harvard standard.* The writer has made a similar reduction, based on incomplete but practically sufficient comparisons of the magnitudes of the first two volumes of the Cordoba Durchmusterung to the Harvard standard as defined by the southern Harvard photometry.f A desideratum is the extension of the Harvard photometry to a sufficient number of the faintest stars admitting of photo- metric determination, with a view of carrying the determination of the values of r^ up to the highest obtainable value of m. 10. The investigations of Kapteyn J on the Luminosity of the Stars which lead to an estimate of their statistical distribution according to absolute luminosity, and of de Sitter § on the systematic difference, depending on the galactic latitude, between photographic and visual magnitude. 11. When the law of progression of the series is determined, an estimate of the total amount of light received from all the stars in a given region of the sky becomes possible. We have then to compare this with the actual amount of light so received from darkest and purest sky. A begin- ning in this direction is found in a rude attempt by the writer to determine the brightness of the sky, leading to the conclusion that the amount of light received from a circular degree is, near the pole of the galaxy, 0.9 that of a star of magni- tude 5.0; and in the brightest regions of the galactic agglomerations, about 2.3 times as much. || It seems extremely desirable that this determination should be repeated with better photometric appliances, especially as it seems very difficult to reconcile it with any of the material cited above, except that of Pickering. Rude though it is, it seems to justify the following conclusions : a. In the region of the galactic poles the amount of light received is markedly greater than that from the entire series of stars above defined. |8. In the galactic agglomerations the value of the product / r^ must diminish rapidly after the point /» = 1 1 . * Ueber die Grossenclassen der telescopischen Sterne der Bonner Durchmusterungen, Munchen Sitzungsberichte, Bd. XXVIII, 1898. t Astronomical Journal, Vol. XXII. t Publications of the Astronomical Laboratory at Groningen. No. 11, 1902. §Ibid, No. 2. II Astrophysical Journal, XIV, p. 297.