IRLF 
 
 SB 35 Mfi3 
 
;K 
 
 ; 
 
 A MECHANICAL THEORY 
 
 OF THE 
 
 *-, 
 
 SOLAE COEOKA. : 
 
 BY J. M/SCHAEBERLE. 
 
 i X 1 
 
 ^ 
 
 
 ** \J 
 
 .S 
 
 # ^ 
 
 \ 
 
 SACRAMENTO: 
 
 STATE OFFICE, : : : A. J. JOHNSTON, SUPT. STATE PRINTING. 
 
 1891. 
 

I'late VII. 
 
 Models illustrating the method, of" fdhntlorfol the 
 
 Polar Ray<3.! ; \\i 
 

A MECHANICAL THEORY 
 
 OF THE 
 
 SOLAE COKONA. 
 
 BY J. M. SCHAEBERLE. 
 H 
 
 SACRAMENTO: 
 
 STATE OFFICE, : : A. J. JOHNSTON, SUPT. STATE PRINTING. 
 
 1891. 
 
S4- 
 
A MECHANICAL THEORY OF HfE < 3Q.LAK 
 
 By J. M. SCHAEBERLE. 
 
 The essential principles which lie at the foundation of the 
 theory here advanced were first published last spring in a short 
 note entitled "A Mechanical Theory of the Solar Corona." 
 (Publications, A. S. P., No, 7; Mon. Not., R. A. S., Vol. 4, p. 
 372.) One year previous to this time (in February, 1889), I 
 expressed my belief to Professor HOLDEN and other astrono- 
 mers at this Observatory, that the coronal form was produced 
 by streams of matter ejected from the lower latitudes of the 
 Sun. At the time Professor HOLDEN desired me to work up the 
 subject, and offered to give me any assistance in his power; 
 but as I had not given much attention to solar observation, 
 and was still less familiar with the physical researches of those 
 who made the Sun a special study, I did not deem the subject 
 a proper one for me to cope with at that time, although I 
 intended to attack the problem later on. 
 
 Through the liberality of Colonel CROCKER, who offered to 
 bear the expense of an expedition to South America, to observe 
 the solar eclipse of December 21-22, this problem of the solar 
 corona was to rivet my attention sooner than I had anticipated. 
 
 The highly satisfactory results secured by the Lick Observa- 
 tory party, consisting of S. W. BURNHAM and J. M. SCHAEBERLE, 
 at Cayenne, South America, were such that after the develop- 
 ment of our plates the views which I had held for nearly a year 
 were very much strengthened. 
 
 By the time we again reached the Lick Observatory the super- 
 structure of the present theory was built, and it only required 
 the finishing touches relating to the more minute details of the 
 corona, and a comparison with published observations, to bring 
 the matter to its present shape. 
 
 In what follows I shall present the theoretical investigations 
 in as simple and brief a manner as possible. 
 
 736351 
 
2 A Mechanical Theory of the Solar Corona, 
 
 $ 
 
 Three well known facts serve as a basis on which the whole 
 theory rests. They are: 
 
 1. The eruptions of the Sun's surface are most active and 
 
 .' The Sun 'rotates, apout an axis passing through its center. 
 \ ; &< \Ti$i$aZis i\ Helped to the plane of the Earth's orbit at an 
 angle of about 82%. 
 
 As the present theory also enables one to reproduce, graph- 
 ically, practically all the phenomena observed during a total 
 eclipse of the Sun, and as the graphical construction is in strict 
 accordance with known mechanical laws, the following brief 
 statement of the theory may be properly made here: 
 
 The theoretical corona is caused by light emitted and reflected 
 from streams of matter ejected from the Sun, by forces which, in 
 general, act along lines normal to the surface of the Sun ; these 
 forces are most active near the center of each Sun-spot zone. 
 
 Owing to the rotation of the Sun, the streams of matter will 
 not lie along normals, since the angular velocity of different 
 portions of a stream grows less as the distance from the Sun 
 increases; in other words, the streams are of double curvature. 
 Each individual particle of the stream, however, describes a 
 portion of a conic section, which is a very elongated ellipse, so 
 long as the initial velocity is less than three hundred and 
 eighty-three miles per second (assuming that the Sun's atmos- 
 phere, as shown by various observations, is exceedingly rare). 
 
 Certain variations in the type of the corona admit of an ex- 
 ceedingly simple explanation, being principally due to the 
 change in the position of the observer with reference to the 
 plane of the Sun's equator. According as the observer is above, 
 below, or in the plane of the Sun's equator, the perspective over- 
 lapping and interlacing of the streamers cause the observed 
 apparent variations in the type of the corona. 
 
 The general direction in which these ejective forces act will 
 be along radii. There may, of course, be numerous particular 
 exceptions to this general direction. 
 
 Similarly, since observation shows that certain solar motions 
 are apparently confined to special regions, which are symmet- 
 rically situated with reference to the axis of rotation of the Sun, 
 we are evidently justified in assuming that the forces which 
 cause these motions are, in general, uniformly distributed 
 
By J. M. Schaeberle. 3 
 
 around each circle of latitude, since there is no a priori reason 
 why, in a highly heated rotating mass of matter, the solar ac- 
 tivity should be confined to any particular circles of longitude. 
 There may, of course, be particular exceptions to a uniform dis- 
 tribution of the forces. 
 
 As the exact nature of the atmosphere surrounding the Sun 
 is unknown, and as its extreme rarity has been demonstrated 
 by various observations,* the heliocentric motions of all parti- 
 cles exterior to the Sun's visible surface will be considered to 
 be unimpeded by solar atmospheric resistances. 
 
 What precedes may be taken to be a complete statement of 
 the general features of the mechanical theory of the corona. 
 In order to compare it with observation it will be necessary to 
 
 *NOTES. With reference to the views of solar physicists on the subject of the 
 Sun's atmosphere, I quote a few paragraphs giving the most recent conclu- 
 sions : 
 
 Professor YOUNG, in his General Astronomy, says: "The corona cannot be a 
 true ' solar atmosphere ' in any strict sense of the word. No gaseous envelope 
 in any way analogous to the earth's atmosphere could possibly exist there in 
 gravitational equilibrium under the solar conditions of pressure and tempera- 
 ture. The corona is probably a phenomenon due somehow to the intense 
 activity of the forces there at work; meteoric matter, cometic matter, matter 
 ejected from the Sun, are all concerned. 
 
 " That this matter is inconceivably rare is evident from the fact that in sev- 
 eral cases comets have passed directly through the corona without experiencing 
 the least perceptible disturbance of their motions. It is altogether probable 
 that at a very few thousand miles above the Sun's surface, its density becomes 
 far less than that of the best vacuum we can make in an electric lamp." 
 
 The following paragraphs are from Mr. KEELER'S Report on the Solar Eclipse 
 of January 1, 1889 (published by the Lick Observatory): 
 
 " Spectroscopic observation furnishes us with many facts which cannot be 
 reconciled with the theory of an extensive solar atmosphere. Two of the most 
 important objections drawn from this source are given below; the first has 
 long been recognized as especially perplexing: 
 
 " (1) The pressure at the surface of the chromosphere is shown by the spec- 
 troscope not to exceed that of a few millimetres of mercury, and this we must 
 accept as the pressure due to an atmosphere from half a million to a million 
 of miles deep, notwithstanding the fact that the force of gravity is twenty- 
 seven times as great at the surface of the Sun as at the surface of the Earth. 
 
 "(2) According to theory, as well as observation, the upper limits of the 
 gaseous envelopes of the Sun ought to be ordered according to their densities. 
 The material which produces the 1474 K line, and which may always be seen 
 in the chromosphere spectrum, is, according to this criterion, as unmistakably 
 denser than hydrogen as is magnesium vapor or iron vapor; but if we accept 
 the coronal spectrum as evidence of the existence of an atmosphere, we are, 
 by exactly the same principle, driven to the same conclusion that the 1474 K 
 material is far less dense than hydrogen. The contradiction could not be 
 more abrupt and inexplicable." HASTINGS. 
 9 s 
 
4 A Mechanical Theory of the Solar Corona, 
 
 deduce the laws which govern the appearances of a typical 
 corona produced by solar eruptions of the character already de- 
 scribed. For convenience, I have made no distinction between 
 the real and these theoretical solar phenomena; it should 
 therefore be borne in mind that in the discussion for deducing 
 the laws of coronal phenomena this theoretical corona is always 
 the one referred to. 
 
 We shall first inquire into the form of the trajectory of a 
 particle ejected from the Sun by a force sufficient to remove it 
 one or more diameters from the Sun's surface. 
 
 If there were no motion of rotation, all particles ejected from 
 a given point on the Sun's surface would, in general, lie along a 
 normal passing through this point. The rotation of the Sun, 
 however, introduces other forces, the resultant of which can, 
 without appreciable error, be regarded as acting at right angles 
 to both the axis of rotation and the line along which the 
 ejective force acts. 
 
 In considering the motion of an ejected particle: 
 
 Let T denote the time of one rotation of the Sun (in latitude <p). 
 Let #0 denote the radius of the Sun. 
 
 Let R denote the radius-vector of the ejected particle at the time t. 
 Let ff> denote the heliocentric latitude of the point of ejection. 
 Let v : denote the linear velocity of the particle due to the Sun's rotation. 
 Let v 2 denote the velocity of ejection. 
 Let V denote the resultant initial velocity of the particle. 
 Let ^ denote the angle between a normal at the point of ejection and the 
 actual initial direction of motion. 
 
 The following expressions give the relations existing between 
 the quantities which determine the subsequent motion: 
 
 If T is expressed in seconds, v l is the distance described in 
 one second of time. 
 
 v\ (2) 
 
 l (3) 
 
 It is evident that so long as the line of action of the ejective 
 force is along a normal, any transverse force, however great or 
 small, will at once determine the position of the plane in which 
 
By J. M. Schaeberle. 
 
 the subsequent motion of the particle lies, since the lines of 
 action of both forces must lie in this plane. If there are sev- 
 eral transverse forces, the plane of motion must contain a 
 normal through the point of ejection and the resultant of the 
 several transverse forces. 
 
 The value of v x is dependent only on q>, and can therefore be 
 determined accurately. If we assign different values to v 2 we 
 can evidently compute all the elements of the several orbits 
 which would be described corresponding to different initial 
 velocities, since we also know the mass and radius of the Sun. 
 In other words, we have given the radius- vector and the veloc- 
 ity and direction of motion of a particle at a given instant of 
 time to find all the elements of the orbit described, the central 
 force being known. 
 
 Let a denote the semi-major axis of the eclipse described. 
 Let X 2 denote the mass of the Sun. 
 
 (The value of K' 2 is such that if the Sun acts on a stationary 
 material particle free to move, at a distance equal to the Earth's 
 mean distance from the Sun, the velocity of the particle at the 
 
 end of one second of time will be equal to 24x60x60 ) 
 The general equation for undisturbed motion is: 
 
 From which we can evidently find a. 
 
 We next obtain the periodic time t from the familiar expres- 
 sion: 
 
 *==-[- (5) 
 
 The eccentricity s can be found as follows: 
 
 Let R and R' denote the distances of the particle from the 
 two foci of the eclipse, then for R R the angle included be- 
 tween R and R is always equal to 180 2^, so that we can at 
 once write: 
 
 s = V R 2 + R 2 2 R R cos (180 2 #) (6) 
 
 from which s is readily found. 
 
 To determine the inclination of the plane of the orbit to the 
 Sun's equator, we have given a point (the Sun's center) and a 
 line (the actual path of the particle the projection of this path 
 on the Sun's surface at the instant of ejection being nearly parallel 
 to the equator) through which the plane of the orbit must pass. 
 
A Mechanical Theory of the Solar Corona, 
 
 The inclination of the plane must evidently always be nearly 
 
 the same as the latitude of the point where the particle is ejected. 
 
 I have computed the paths of particles for different velocities 
 
 of ejection in a mean latitude of 15. The elements are given 
 
 in the following table: 
 
 TABLE I. 
 
 Initial Velocity 
 of an Ejected 
 Particle=V. 
 
 Angle between 
 the Normal at 
 the Point of 
 
 Semi-Major Axis 
 of the Orbit= 
 
 Periodic Time 
 of the Parti- 
 cleif 
 
 
 
 
 Ejection and 
 
 
 
 
 Miles per second. 
 
 the Initial Di- 
 rection of Mo- 
 tion;^. 
 
 Radius of Sun=l 
 
 Expressed in 
 Days. 
 
 Eccentricity. 
 
 219.1 
 
 0.318 
 
 0.75 
 
 0.075 
 
 0.9933 
 
 269.2 
 
 .259 
 
 1.00 
 
 .116 
 
 .9950 
 
 311.3 
 
 .225 
 
 1.50 
 
 .214 
 
 .9967 
 
 330.3 
 
 .212 
 
 2.00 
 
 .329 
 
 .9975 
 
 348.3 
 
 .201 
 
 3.00 
 
 .604 
 
 .9983 
 
 357.0 
 
 .196 
 
 4.00 
 
 .930 
 
 .9987 
 
 362.1 
 
 .193 
 
 5.00 
 
 1.300 
 
 .9989 
 
 372.2 
 
 .188 
 
 10.00 
 
 3.678 
 
 .9995 
 
 377.0 
 
 .185 
 
 20.00 
 
 8.263 
 
 .9997 
 
 379.5 
 
 .185 
 
 40.00 
 
 29.42 
 
 .999+ 
 
 379.7 
 
 .185 
 
 80.00 
 
 83.22 
 
 .999+ 
 
 382.0 
 
 .184 
 
 Infinity 
 
 Infinity 
 
 1.000 
 
 Apparent motions corresponding to component velocities of 
 two hundred and fifty miles per second (the actual velocity is, 
 as a rule, always greater than the observed) are not infre- 
 quently observed in solar protuberances. (See Young's General 
 Astronomy, page 208.) It does not, however, follow that the 
 forces producing these motions must be correspondingly great, 
 for if: 
 
 (in which F, V, and M are respectively the moving force, the 
 maximum velocity generated, and the mass moved), then for a 
 constant force F the velocity V will increase as the mass M 
 diminishes, and as the resistance to motion is to be considered 
 zero, the volume corresponding to a given mass may be either 
 great or small without affecting the velocity. 
 
 The above elements show that the theoretical orbit will 
 always be a very elongated ellipse, the perihelion point being 
 near the Sun's center and having a longitude nearly 180 dif- 
 ferent from the longitude of the point of ejection, while the 
 nodal points will always be 90 and 270 from the theoretical 
 perihelion, and within less than a score of miles of the Sun's 
 center. 
 
By J. M. Schaeberle. 
 
 Let us now consider the circumstances of motion of a series of 
 particles ejected from the same point on the Sun's surface following 
 each other so as to form a continuous stream. The angular velocity 
 of each particle as it leaves the Sun's surface will be the same 
 as that of the Sun, but as the radius-vector of the particle 
 increases the angular velocity decreases according to rigorous 
 mechanical laws. Therefore, while each particle may be con- 
 sidered as describing an independent orbit, the curve formed 
 by the stream will not be a conic section, but a helix, which, 
 so long as the latitude of the base of the stream is not zero, 
 will necessarily be of double curvature. As all the particles 
 causing the visible coronal outlines are within a few diameters 
 of the Sun, we can, without appreciable error, consider a given 
 stream as lying in one plane. For simplicity in the discussion, 
 a parabolic velocity will be assumed for all streams: 
 
 Let &)o denote the true anomaly for a radius-vector 7? and time to. 
 
 Let &> denote the trae anomaly for a radius- vector R and time t. 
 
 Let T denote time of the theoretical perihelion passage. 
 
 Let t denote the time corresponding to R and GO. 
 
 Let denote the angle turned through by the Sun during the time (t to). 
 
 With the well known relations which exist between (t T ) and 
 oo in the case of parabolic motion, the results given below are 
 readily obtained. The angle 6 is zero at the instant t when 
 GO = GO O and R = R . Both 6 and GO GO O are measured from the 
 fixed heliocentric direction in space which the origin of the 
 stream has at the instant . The angle which a line drawn 
 from the base of the stream to any point in the stream makes 
 with a normal through this base, will be called /. 
 
 TABLE II. 
 
 Radius-Vector of 
 the Particle=-ff= 
 
 1.0. 
 
 2.0. 
 
 3.0. 
 
 4.0. 
 
 5.0. 
 
 6.0. 
 
 180 oo 
 
 22M 
 
 15'. 6 
 
 12'. 8 
 
 o ir.o 
 
 9'.9 
 
 (yy.o 
 
 tT 
 
 O d .009 
 
 O d .025 
 
 Oi.045 
 
 l^.OTO 
 
 O d .097 
 
 (H.128 
 
 t 1 
 
 O d .OOO 
 
 Od.016 
 
 O d .036 
 
 O d .061 
 
 Od.088 
 
 O d .119 
 
 &) a? . 
 
 O'.O 
 
 6'.5 
 
 V.S 
 
 ll'.l 
 
 12'. 2 
 
 13M 
 
 
 
 o.o 
 
 0.2 
 
 0.5 
 
 0.8 
 
 1.2 
 
 1.6 
 
 J 
 
 
 0.2 
 
 0.5 
 
 0.9 
 
 1.3 
 
 1.8 
 
 
 
 
 
 
 
 
8 A Mechanical Theory of the Solar Corona, 
 
 TABLE II Continued. 
 
 Radius- Vector of the Parti - 
 
 016=^2= 
 
 7.0. 
 
 8.0. 
 
 10.0. 
 
 20.0. 
 
 40.0. 
 
 180 GO 
 
 8'.3 
 
 7'. 8 
 
 7'.0 
 
 4'. 9 
 
 3' 5 
 
 tT 
 
 O d .162 
 
 0* 198 
 
 O d .278 
 
 O d .782 
 
 2 d 18 
 
 tt ._. 
 
 O d .153 
 
 O d .189 
 
 O d .269 
 
 O d .773 
 
 2 d .17 
 
 G) (i3 _. 
 
 13'. 7 
 
 14'.3 
 
 15'.1 
 
 17'.2 
 
 18'.6 
 
 e 
 
 2 1 
 
 2 6 
 
 
 
 
 
 2.3 
 
 2. 8 
 
 
 
 
 
 
 
 
 
 
 The equation which fairly represents the relation between R 
 and /, I find to be the following: 
 
 I = j,IF + tX-l, (7) 
 
 / being expressed in degrees of arc and the unit of R being a 
 radius of the Sun. 
 
 For discussing the curve represented by the above equation, 
 it will be more convenient to take the origin on the Sun's sur- 
 face at the base of the stream and refer the curve to a normal 
 through this origin. 
 
 For all points in any stream visible during an eclipse we can 
 without sensible error write: 
 
 P = R 1, (8) 
 
 p being the distance from any point in the stream to the Sun's 
 surface. 
 
 Equation (7) now becomes 
 
 f='*(P + !)' + * A (9). 
 
 Since for our limits the angle I will always be small, the in- 
 clination r of the curve, at any point, to a normal through the 
 origin will be given by the differential expression 
 
 ^ = Pgj5-+J=*p(p-fl)+tP + I, (10) 
 
 P being expressed in degrees of arc. 
 
 With the aid of this equation we obtain the following incli- 
 nations P for the corresponding values of p, the values of / 
 being taken from Table II: 
 
By J. M. Schaeberle. 
 
 TABLE III. 
 
 1.0 
 
 0.5 
 
 2.0 
 
 1.2 
 
 3.0 
 
 . 2.0 
 
 4.0 
 
 3.0 
 
 5.0 
 
 4.1 
 
 6.0 
 
 5.4 
 
 7.0 
 
 6.8 
 
 If ft denotes the Earth's angular distance above or below the 
 plane of the Sun's equator at the time of observation, a streamer 
 in latitude <p, in order to be just visible at either pole of the Sun, 
 must have a length p given by the equation: 
 
 Po= 
 
 sn c 
 
 1 
 
 (11) 
 
 The values of p for the two latitudes 15 and 30, and for 
 particular values of ^, are given in the following table (re- 
 ferred to the nearer hemisphere) : 
 
 TABLE IV. 
 
 Latitude on the =<p. 
 
 Angular Distance of the 
 Earth above the Plane of 
 the 0's Equator =/?. 
 
 Length of a Streamer from 
 the Sun, which will just 
 be Visible at the 's 
 Pole=/-V 
 
 -4-15 
 
 + 7.25 
 
 1.64 
 
 +15 
 
 + 7.25 
 
 6.42 
 
 +15 
 
 0. 
 
 2.86 
 
 +30 
 
 0. 
 
 1.00 
 
 +30 
 
 + 7.25 
 
 0.65 
 
 +30 
 
 + 7.25 
 
 1.59 
 
 A simple way to represent graphically the apparent changes 
 in the form and position of a streamer, due to the change in 
 the position of the Earth and to the rotation of the Sun, is to 
 construct a model of a single streamer issuing from a sphere 
 and to project the streamer on a plane surface by parallel rays 
 of light. If the sphere be rotated, and the inclination of the 
 axis of rotation varied between the proper limits (correspond- 
 ing to the positions of the Earth at different seasons of the 
 year), all the peculiarities of the appearances of the coronal 
 streamer can be reproduced and studied at leisure. 
 
 In order to show more forcibly the varying form and position 
 
10 A Mechanical Theory of the Solar Corona, 
 
 of a streamer in projection, I have exaggerated the curvature in 
 the model used for obtaining the outline given in Plate VIII. 
 
 The streamer represented by a wire rod was made to issue 
 from a latitude of about 15, and to have such a length that in 
 projection it was just visible at the pole for j3 = 0. A longer 
 streamer would have necessitated the use of a smaller scale. 
 
 The figures at the extremities of the lines representing the 
 streamers correspond to the number of days elapsed since the 
 base of the streamer was on the central meridian of the nearer 
 hemisphere. 
 
 Fig. 1, of Plate VIII., corresponds to an inclination ft = 0, 
 Fig. 2 to the time when the Earth is near its maximum dis- 
 tance below the plane of the Sun's equator, and Fig. 3 to the 
 time when the Earth is near its maximum distance above the 
 same plane. 
 
 The general outlines of the figures on Plate VIII. are evidently 
 the same as would be produced by a series of equidistant 
 streamers issuing from a single latitude only. 
 
 If now the streamers issue from different latitudes within 
 the limits of the Sun-spot zones, coronal forms similar to those 
 shown on Plate VI. (which are made from an actual model, as 
 described on the plate itself ) will be produced. In a typical 
 corona the mean latitude of the zone from which the streamers issue 
 will always be the same as the latitude of the densest and most 
 extended portion of the coronal outlines. A necessary consequence 
 of the perspective overlapping of these streamers is the production 
 of luminous rays which, in a typical corona, incline away from the 
 adjacent poles, as will be demonstrated further on. 
 
 A crucial test of this or any other theory of the corona is 
 afforded by the appearance and behavior of these so called 
 "polar rays" If they really originate at or very near the Sun's 
 polar regions, the laws governing their appearance must be 
 sought for in the physics of the Sun itself. If, however, the 
 present conception of their origin is the true one, they are 
 mainly due to a series of apparent intersections of streamers 
 from the Sun's equatorial region, projected by perspective above 
 or below the poles; that is, the polar rays which we see in any 
 corona have no objective existence. The situation and curva- 
 ture of these rays and the apparently vacant spaces where such 
 rays seem to be wanting (the so called "rifts") can be accu- 
 
PLATE VIII. 
 
 DIAGRAM ILLUSTRATING VARIATION IN THE CURVATURE AND POSITION 
 OF A SINGLE STREAMER IN PROJECTION, BY J. M. SCHAEBERLE. 
 
By J. M. Schaeberle. 11 
 
 rately observed, and the present theory enables one to account 
 for such rays and rifts, and to give the inclinations and general 
 directions of their curvatures. 
 
 We will now proceed to investigate the curvatures and situa- 
 tions of the polar rays, first deducing the expressions which 
 give, in projection, the positions and curvatures of the streamers. 
 
 I call a "streamer" the actual stream of particles ejected 
 from the Sun. I call a "ray" the subjective appearance pro- 
 duced by overlapping streamers seen in projection, and I write 
 "ray" in italics. 
 
 We will first consider the problem for the case when fi=Q. 
 Let T denote the heliocentric longitude of the base of a streamer 
 in latitude cp, and let the longitude of the Earth be r , the plane 
 of the Sun's equator being taken as the fundamental plane. A 
 normal in latitude cp and longitude t r will in projection inter- 
 sect the Sun's limb at a polar distance p given by the equation: 
 
 (12) 
 
 tan cp 
 
 The position angle p f of the inclined stream for moderate polar 
 distances can then be found from the approximate expression: 
 
 cos (* T Q) (13) 
 
 The inclination ( J") of the stream to a normal through its 
 
 base will in projection be given by the approximate expression: 
 
 jr,_/cos(rr ) 
 
 Finally, the inclination T to a normal passing through the 
 point at which the inclination is required, will be approxi- 
 mately: 
 
 r=r-(p-p') (15) 
 
 Now, owing to the inclinations of the streamers, it is at once 
 evident that even when they are uniformly distributed in longi- 
 tude, they can never be symmetrically arranged around a radius- 
 vector of the Earth. In projection, however, an approach to 
 a symmetrical form results from a peculiar but well known 
 optical phenomenon. 
 
 The streamers which have the least inclination to the line of 
 sight will evidently appear to be most widely separated; that 
 is, they will always be at the areas of minimum density. For 
 the nearer hemisphere the minimum density in projection will 
 
12 A Mechanical Theory of the Solar Corona, 
 
 evidently always be just a little to the west of the Sun's pole, 
 while for the farther hemisphere it will be to the east of the 
 Sun's pole. 
 
 Again, as the streamers are slightly curved, those having the 
 least inclination to the line of sight will in projection appar- 
 ently have the greatest inclination with reference to normals. 
 Hence, when the Earth is near either node, an approximately 
 uniform distribution in longitude of streamers from a given 
 latitude cp will result in the following arrangement in ortho- 
 graphic projection: 
 
 As streamers issuing from the nearer hemisphere have an 
 eastward inclination with reference to normals, the two areas 
 of minimum density will be a little to the west of the Sun's 
 poles; while for the farther hemisphere as the streamers have 
 a westward inclination with reference to normals, the corre- 
 sponding areas of minimum density will be a little to the east 
 of the Sun's poles. 
 
 The inclination to the normal is greatest near the points of 
 minimum density, and this inclination gradually grows less 
 and the density greater as the angular distance from these 
 points increases, until the polar distance 90 cp is reached, 
 where the inclination to the normal is practically zero, and the 
 density at a maximum. ( See illustrations of model, Plate VIII. ) 
 
 The apparent intersections of these streamers produce false 
 streamers, which I have called rays, to distinguish them from 
 the streamers proper. 
 
 The form of these rays for a theoretical corona can be deter- 
 mined after the method which will now be considered. 
 
 THEORY OF THE INCLINED RAYS. 
 
 From a series of experiments I have derived the expressions 
 given below, which govern the formation of certain luminous 
 lines (which for want of a specific name I shall simply call 
 rays) produced, as is well known, by the overlapping of two 
 or more sets of alternate bright and dark lines. 
 
 If a set of parallel luminous lines, separated by the small 
 angular distance D, is crossed at the small angle J by a second set 
 of parallel lines separated by the slightly less angular distance D', 
 then when either set is viewed through the other, a new and heavier 
 set of luminous lines (rays) will be formed, which make the angle 
 
By J. M. Schaeberle. 13 
 
 K with the wider set of lines, and the angle K / with the set of 
 closer lines. 
 
 I have deduced the following rigorous expression for the 
 value of K: 
 
 tan g== _? - r tan J (16) 
 
 D DsecJ 
 
 The perpendicular distance D l between two adjacent lines 
 (rays) of the new set is given by the rigorous expression: 
 
 D 1 =D(cot /cot #)sin K (17) 
 
 'Equation (16) shows that K is always greater than J, and 
 reaches a maximum when D D'sec J=0; for this condition 
 #=90,and D l =D cot/. 
 
 As the angle / increases, the number of intersections to a 
 given surface increases. Now as these intersections cause the 
 phenomenon of the inclined rays (a ray passes through those 
 intersections which are nearest together), it is evident that the 
 rays will be least conspicuous when the number is greatest, 
 since the contrast between different parts of a surface covered 
 by such rays diminishes as the intervals grow smaller. 
 
 If one of the two sets of lines is doubled by interpolating a 
 new line midway between each of two adjacent lines, a new ray 
 will be formed midway between each of the original rays. 
 
 Similarly, if both sets of lines be supposed to change in the 
 same way, precisely the same effect will be produced, the num- 
 ber of rays being quadrupled. 
 
 If this process of bisection is continued indefinitely, a limit 
 is soon reached at which the lines and rays form a practically 
 continuous surface. The inclination of these rays always remains 
 parallel to the original direction corresponding to the particu- 
 lar condition D D'=x] the K for this particular series of rays 
 can be distinguished from other series of less conspicuous rays, 
 made by the same set of lines, by writing: 
 
 The values of K for each of the other sets of inconspicuous 
 rays can be found by substituting in (17'), successively, the 
 
 jy jy jy T\> 
 
 values _,_,--_ ._.--^-, in place of D'. 
 
 The most favorable case for conspicuous rays evidently 
 
14 A Mechanical Theory of the Solar Corona, 
 
 results when / is small, and DD'=x (x being a small quan- 
 tity), since for these conditions any two intersecting lines 
 have many points in common, supposing the lines to have 
 sensible breadths, and only one value of K is possible, so that 
 all the rays are inclined in the same direction, thus at once 
 rendering them unmistakable. 
 
 Equation (17) shows that so long as J is very small, the 
 number of rays increases very rapidly as J increases. For 
 large values of J, therefore, no conspicuous rays will be shown. 
 
 As the angle J=2I' will always be small so far as the pres- 
 ent investigation is concerned, we can, without appreciable 
 error, write, in place of equation (16), the expression: 
 
 tan K=-^-p tan / (18) 
 
 I tried to deduce a rigorous expression for the equation of 
 the curve formed by the projection of the nearer set of stream- 
 ers upon the farther set, but found the conditions too compli- 
 cated, since there are two sets of curved diverging lines, neither 
 set having a common point of intersection, and the two sets 
 having different envelopes, while the angle of divergence itself 
 is a variable quantity. 
 
 We can, however, apply the above formula for finding K, if 
 we consider only very short portions of the curve, since for 
 very short distances the deviation from parallelism of two 
 adjacent streamers will be small, and consequently the error in 
 K will, as a rule, be small also. 
 
 Now for a series of normals, radiating at regular intervals 
 from a small circle whose latitude is cp, the angle between the 
 normals in the nearer and farther hemispheres for the same 
 position-angle p is always zero in projection; hence K is always 
 zero, whatever the relative density of the normals in the two 
 hemispheres may be. The projected intervals will not vary as a 
 function of p alone, since for normals in latitude (p, the density 
 will be at a maximum when p=$Q cp; hence it follows that 
 the intervals vary as some function of ap, a being defined by: 
 
 Now, in the case of an actual streamer, when the base is in 
 longitude r T O) the longitude indicated by the direction of the 
 
By J. M. Schaeberle. 15 
 
 stream at a distance p from the base will be (r T O /'); hence, 
 when the direction of the streams, at a distance p, corresponds to 
 T T =0 and T T Q =180, the bases of these streams will be 
 in longitudes -{-I' and 180-+-!', respectively. Now, in projection, 
 the position-angles, p^ and p 2 corresponding to these longitudes, 
 are given by the approximate expressions: 
 
 sn 
 
 In which 1= -$((& & ), p being measured from the north 
 pole towards the east, through 360. 
 
 Pi and p 2 will be called the secondary coronal poles of the 
 nearer and farther hemispheres, respectively, for the latitude (p. 
 
 The location of the secondary poles can perhaps be better 
 understood from what follows. 
 
 At the north pole, for example, streamers from the nearer 
 hemisphere, which are projected on the west side of the second- 
 ary pole, appear to radiate from points which are below the 
 Sun's center. As the position angle increases, the center of 
 divergence appears to approach the Sun's center, and then to 
 rise above it, so that streamers on the east side appear in pro- 
 jection to radiate from points above the Sun's center. 
 
 Consequently, at the same true polar distances, the density 
 on the east side of the Sun will be greater than it is on the 
 west side. For the streamers of the farther hemisphere pro- 
 jected at the same pole, the arrangement is evidently exactly 
 the reverse. 
 
 A streamer is at a secondary pole when its direction in projec- 
 tion is parallel to the Sun's axis. 
 
 Equation (16) can be written: 
 
 tan K= -- ^ tan / (21) 
 
 From which it appears that K is only dependent on the ratio 
 of D' to D, and entirely independent of their absolute values. 
 
 Let us consider / to be positive when the inclination of the 
 denser set of projected streamers to the less dense set is in the 
 
16 A Mechanical Theory of the Solar Corona, 
 
 direction N.E.S.W., and negative, when the inclination is in 
 the opposite direction; then, in conformity with the law stated 
 
 on page 58, the factor: 
 
 D 
 DD' 
 
 will be positive in the first and second quadrants (but inde- 
 terminate between the limits _p=90 cp and 90+^?), and 
 negative in the third and fourth quadrants (but indeterminate 
 between the limits p=27Q(p and 270+^); hence, ac- 
 cording to the above equation, K will always have the same 
 sign as J", or, in other words, the rays in latitudes greater than 
 <p will always be inclined in a direction away from the nearer 
 pole. The angle which the ray makes with a normal will be 
 K fj, but as / is, as a rule, smaller than the error of the 
 measured inclinations, and / always less than 0.5, the term f'J 
 can be neglected. K can therefore be considered to be the 
 inclination of the ray to a normal, and if the quadrant is 
 mentioned no attention need be paid to the sign of K, as the 
 inclination is in the direction away from the nearer pole. 
 
 For finite values the condition DD' will only be fulfilled at 
 the poles where the density is near a minimum; the very short 
 concentric arcs formed by the few intersections of nearly normal 
 streamers inclined to each other at the small angle /, will there- 
 fore not be visible as a rule. The nearly normal rays at the 
 poles, therefore, nearly coincide with the streamers. This par- 
 ticular phase of the phenomenon is beautifully illustrated by 
 means of the models given in Plate VII., described further on. 
 Figure I. illustrates one of the cases which correspond to the 
 the value /?=0, so that in orthographic projection D=D' at 
 the poles; for this exceptional position the slightest variations 
 of the angle J", produced either by elevating one set of lines with 
 respect to the other, or by a rotary motion of the plates, causes 
 the ray to swing very rapidly through a large angle, producing 
 forms like Figures II., III., IV., XV., all corresponding to small 
 values of J. (Figures X. and XI. represent conditions when J 
 is comparatively large and D=D'.) Except for critical cases, 
 a given value of .fiTdoes not change to any great extent for small 
 relative variations of Z), D', /, as can be seen from an inspec- 
 tion of the figures. At the same position-angle p the Value of 
 K will vary but slightly when p is comparatively large, and 
 
By J. M. Schaeberle. 17 
 
 although the apparent changes are very rapid for a given ray, 
 yet as each one assumes a new position another ray is formed 
 which nearly occupies the successive positions of the preceding 
 ray, so that at any given instant the apparent change in the 
 whole system of rays will only result in a lateral displacement 
 of the axis of symmetry, and for the critical cases the concen- 
 tric rays formed are usually not of sufficient magnitude to be 
 seen, as has been explained above. When both sets of stream- 
 ers are visible, the pyramidal form of the denser portions of the 
 rays projected at the poles can be regarded as resulting directly 
 from the overlapping of streamers inclined to each other. 
 
 Now the streamers at the nearer secondary pole are projected 
 against streamers of the farther hemisphere, which are at an 
 angular distance of 21 (=J) from the farther secondary pole; 
 for the polar distance p the ratio of the intervals D and D' is 
 therefore a function of ap and /'. This function must be so deter- 
 mined that the density is least, but not zero, when p=Q I', 
 and a maximum finite density when p=90 (p, or ap=90. 
 
 Let # and d' denote the densities of the two sets of streamers 
 for a given polar distance p. We then have as a rough approx- 
 imation for values of p greater than J : 
 
 d=d sin Op-f /:) 
 d'=d sin (ap I,} 
 
 The value of d being determined by the condition: 
 
 (23) 
 
 In which S is the density at the true pole. 
 
 In place of the density tf, we can, for moderate polar dis- 
 tances, substitute the reciprocal of the corresponding interval 
 D, and write: 
 
 Hence: 
 
 D d' sin (apI,) 
 Substituting this value of the ratio of the intervals in equa 
 tion (21), and reducing, we obtain, finally: 
 
 tan K= - tan 2T (26) 
 
 2cos (ap) sin / 
 
18 A Mechanical Theory of the Solar Corona, 
 
 21' being substituted in place of /. As /' is small we can 
 write tan 2/' 2 tan/'; hence, 
 
 tan K= Bn , a f- l r tan /' (27) 
 
 cos (ap) sin J, 
 
 The ratio of tan J 7 to sin /x will be practically constant for 
 moderate values of p. When ap=SQ, equation (27) becomes 
 indeterminate. If the streamers were of infinitesimal breadth 
 in latitude, they would coincide with the rays when p=9Q q>. 
 In reality, however, the extension in latitude may be consider- 
 able, and the effect of this extension is to make a practically 
 continuous surface of rays before the polar distance is as great 
 as 90 (p. Hence, polar rays at considerable polar distances 
 will usually be lost in the general illumination. 
 
 From the preceding discussion it follows that for the case of 
 ft=Q the axis of the typical corona will, in orthographic pro- 
 jection, coincide with the Sun's axis. 
 
 CHANGES IN THE FORM OF THE TYPICAL CORONA RESULTING 
 FROM VARIATIONS IN THE HELIOCENTRIC LATITUDE OF THE 
 EARTH. . 
 
 When the Earth is above (/?+) or below (/? ) the plane of 
 the Sun's equator, the quantities D, /)', and / undergo certain 
 changes for the same value of p in projection. 
 
 When the Earth is below the plane of the equator, the nearer 
 streamers of the northern hemisphere will appear more dense, 
 with a slightly less inclination to the normal than is the case 
 when /?=0, while the density of the streamers from the farther 
 hemisphere will be diminished; but the increase in the west- 
 ward inclination to the normal will be greater than the dimi- 
 nution of the eastward inclination in the nearer hemisphere. 
 j has therefore increased, but this increase is, in part, at least, 
 
 counteracted by the decrease in the value of the factor ,. 
 
 D D ' 
 
 With reference to the streamers, therefore, K does not neces- 
 sarily change to any great extent. But as the nearer streamers 
 have a less eastward inclination, while the farther ones have a 
 greater westward inclination, the whole system of rays formed 
 will, according to the fundamental principle laid down in the law 
 given on page 58, have a greater eastward inclination. The 
 
By J. M. Schaeberle. 19 
 
 effect of this comm.on change in the inclination of the rays is to 
 cause an apparent shifting of the coronal pole towards the WEST. 
 
 For the same reason the south coronal pole will be shifted 
 towards the EAST. 
 
 When the Earth is above the plane of the equator, the north 
 pole of the corona will be on the east side of the projected posi- 
 tion of the Sun's pole, and the south coronal pole will lie on 
 the west side. 
 
 As the projections seen from the Earth are not strictly ortho- 
 graphic, the parallactic effect has a constant tendency to throw 
 both coronal poles slightly towards the east of the position they 
 would occupy if viewed from an infinite distance. 
 
 Changes in the form of the typical outer corona, due to varia- 
 tions in ft, for streamers of a given magnitude and distribution, 
 can be readily understood, with the aid of the prints and dia- 
 grams given in Plates VI. and VIII., in connection with Tables 
 IV. and V., and also from what follows further on. 
 
 A want of uniformity in the distribution of the streamers in 
 longitudes r r Q = 0, or 180, may result in the partial or 
 total destruction of the symmetrical inclined polar rays, and in 
 other longitudes the same irregularities will cause a deforma- 
 tion of the general typical features of the corona. An unsym- 
 metrical corona is, therefore, in no way to be regarded as 
 contrary to the theory. 
 
 If, for any cause, there is real or apparent periodic variation 
 in the mean latitude of the streamers, the ratio of the polar to 
 the equatorial extension will, of course, be subject to a similar 
 variation. (See postscript.) 
 
 From Table IV., we see that for two streamers in latitude 15 
 and 30, respectively, just visible at the poles in projection, the 
 one in latitude 30 will have only about one third of the 
 equatorial extension that is shown by streamers in 15 latitude. 
 If, therefore, the mean latitude of greatest solar activity is 
 apparently shifted towards the higher regions of the Sun-spot 
 zones, the coronal outlines may become practically circular. 
 The criterion for determining from the observed outlines 
 whether ft is large or small will no longer be satisfactory, since 
 the ratio of the polar to the equatorial extension now under- 
 goes but a slight change for all possible values of ft. However, 
 when the polar rays, or the wing boundaries adjacent to the poles, 
 3 s 
 
20 A Mechanical Theory of the Solar Corona, 
 
 are present, the position of the axis of the inner corona will, as 
 a rule, at once tell whether the observer was above, below, or 
 in the plane of the Sun's equator. The inclinations of the 
 boundaries of the wings, at the Sun's outline, can be used to 
 indicate, approximately, the position of the pole of the inner 
 corona when the rays are wanting. 
 
 Now, when the Earth is at considerable distances above or 
 below the pl'ane of the Sun's equator, and the equatorial stream- 
 ers are visible for only a moderate distance, the more depressed 
 set may show so faintly that they apparently fall short of reach- 
 ing the pole in projection. For this case, equation (16) shows 
 that K=0, since 21' is then zero; so that if all the streamers 
 were confined to a single circle of latitude and only one set vis- 
 ible at the pole, the true projections of the streamers would be 
 seen, and the direction of curvature would always be governed 
 by rigorous conditions. (See Plate VIII.) 
 
 For this case, when the symmetrical inclined rays are present, 
 the phenomenon admits of the same simple explanation, which 
 accounts for the polar rays when fi is nearly zero, since stream- 
 ers on opposite sides of the Sun, whose latitudes differ by 2ft, 
 will, in projection at the poles, produce a similar arrangement 
 of the rays; but these rays will not necessarily extend to the 
 outer limit of the coronal structure at the poles, and for this 
 reason they may not be conspicuous when ft is large. 
 
 When, however, the polar rays are very conspicuous and 
 uniform for large values of /3, and a small equatorial extension, 
 the phenomenon is still to be accounted for in the same general 
 way, which we will now consider. 
 
 As the curvature and deviation from the normal depend in 
 projection upon the latitude q>, streamers in different latitudes 
 with nearly the same longitude will, in projection, apparently 
 intersect each other, and consequently curved rays or outlines 
 will be formed, and these may be wholly produced in either the 
 nearer or farther set of streamers. For the same position-angle 
 p, streamers in latitude cp' will not appear to radiate from the 
 same points that those in latitude cp do. If the secondary poles 
 for latitude cp' are given by the expressions: 
 
 p i ' = /! ' nearer hemisphere, fr>< -.. 
 
 p 2 ' = + // farther hemisphere, 
 
 Then for -f cp' greater than -f- cp we would always have // less 
 than /!. This results directly from the fact that the change in 
 
By J. M. Schaeberle. 21 
 
 the directions of the streams varies nearly with p 2 , and in pro- 
 jection at the poles the ratio of p to p' is nearly the same as the 
 ratio of cosec cp to cosec qj '. The streamers in latitude (p will 
 therefore be more inclined to the normal than those in latitude 
 (p'. In projection, therefore, streamers on the east side of the 
 secondary pole in latitude (p will appear to radiate from points 
 which are nearer than the corresponding points for latitude cp', 
 while on the west side of the Sun the points on the osculating 
 curves are at greater distances for latitude cp than they are for 
 the latitude cp' . The resulting apparent variation in density will 
 therefore produce the same phenomenon which results from the pro- 
 jection of the streamers of the nearer hemisphere upon those of the 
 farther. Inclined polar rays may therefore be formed by the over- 
 lapping of streamers having nearly the same longitude but differ- 
 ent latitudes. For this case the /j of equation (24) would be 
 the angular distance between the secondary poles for latitude 
 cp and cp', while /' would correspond to the difference of the pro- 
 jected inclinations at the pole for the given latitudes. 
 
 If we assume the ratio of the latter to the former to be c, 
 and neglect the small angle (I I 7/), equation (27) reduces to: 
 
 tan K=c tan (ap') (29) 
 
 In which p' is now the distance from the mean secondary pole. 
 This system of rays will be more probable for large values of 
 /?, and will not be symmetrical with reference to the poles of 
 the Sun. The inclinations will vary with the value of c; these 
 variations will, however, as a rule, be small, since the values 
 of /?, cp, and cp' are subject to comparatively small variations. 
 
 When ft is zero, the streams of the nearer hemisphere pro- 
 jected against those of the farther hemisphere will produce a 
 corona whose axis will be practically coincident or parallel with 
 the Sun's axis. The density (in projection) at the poles may at 
 times be so great that only the wings are seen; the boundary of 
 these wings, even at the poles, will then extend to a limit at 
 which the polar rays begin. 
 
 Equation (21) shows that the inclination of the rays is inde- 
 pendent of the magnitude of the intervals, so that even when 
 the density is very great or very small, the inclination K of the 
 polar rays, when these are present, will, for a given value of ft, 
 always be practically the same for a uniform distribution of the 
 streamers. 
 
22 A Mechanical Theory of the Solar Corona, 
 
 The four great curves which bound the outer portions of the 
 wing-like coronal outlines, which form a prominent feature of so 
 many eclipses, are the envelopes of the outer portions of the 
 streamers, the density at points on the envelopes being such 
 that the intervals are just filled out. The positions of these 
 envelopes are largely determined by the projection of the more 
 depressed set of streamers against the more elevated set, or 
 vice versa. (See Plate VI.) 
 
 To obtain an approximate analytical expression for these 
 envelopes, let y-f be the distance from the Sun's center to the 
 point on the Sun's axis, where one set of streamers in latitude 
 q>, produced backward, would, in projection, just form a con- 
 tinuous surface, and let P f be the polar distance of the point 
 where the boundary of this continuous surface cuts the limb of 
 the Sun in projection. Then the distance y', at which the inter- 
 vals between adjacent streamers will just be filled out for differ- 
 ent values of p, will be approximately given by the expression: 
 
 Y 7i sec (ap) (30) 
 
 In which yf is a function of the actual density, and equal to 
 the distance from the Sun's center to the point where the 
 envelope proper, or the envelope produced, cuts the Sun's axis. 
 
 For values of y', less than the Sun's radius, the approxi- 
 mate expression (31) will answer: 
 
 7/1 = " sec (a**) (31) 
 
 For the envelope of the other set of streamers, projected at 
 the same pole, we have: 
 
 y"=y" sec (ap) (32) 
 
 Similarly, streamers in latitude ?/, cp", etc., will have envel- 
 opes of the same general character. The overlapping of all 
 these envelopes finally produces the outlines of the four great 
 wings. 
 
 The principal boundary curve or curves will be approx- 
 imately represented by the equation: 
 
 r =y lS ec(ap) (34) 
 
 The value of y l being found from data taken directly from 
 the photographs. 
 
By J. M. Schaeberle. 23 
 
 Equation (34) is the locus of a curve of indefinite extent, 
 which leaves the Sun's surface at the polar distance P, or cuts the 
 Sun's axis at a distance y from the Sun's center, the inclination 
 to. the normal varying from 90 for p=Q to for p=90 q>. 
 The curve has for its asymptote a produced diameter of the 
 Sun in a position-angle nearly equal to 90 cp , in which q) m 
 is the mean latitude of the streamers. 
 
 If the distribution of the streamers in longitude is not uni- 
 form, equation (34) will not necessarily represent the observed 
 form for small values of p, and if the terrestrial atmospheric 
 conditions are unfavorable for showing the outer corona, the 
 same equation will apparently fail when p is large. 
 
 The inner boundaries of these wings will be determined by 
 similar conditions. As the forces at the equator are presum- 
 ably less active than those having a small latitude, the envel- 
 ope of these portions of the projected streams will in general be 
 concave near the Sun's disk. The closing of the wings (or 
 fish tails as they are sometimes called) and the decrease in 
 density of the exterior boundaries, result directly from a grad- 
 ual increase in the value of ft. 
 
 The trumpet-shaped outline of the more distant portions of 
 the wings which were first photographed in January, 1889, is 
 theoretically just what is called for. Since the brightness of 
 the streamers decreases with increasing distances from the 
 Sun's surface, the illumination can, by contrast, only be of 
 indefinite extent in those directions where many of these stream- 
 ers overlap each other, that is, in position-angles corresponding 
 to (90 -<p) and (270 <p). The equation of the outer 
 boundary curve is nearly of the form expressed by formula 
 (34). 
 
 The figures given in Plate VI. would be still more strikingly 
 similar to coronal outlines were it practicable to represent the 
 decrease in the brightness of the streamers (represented by the 
 needles in the model) with increasing distances from the center. 
 
 POLAR GAPS OR RIFTS. 
 
 There are several different causes which unite to produce an 
 apparent extinction of the polar rays for increasing polar dis- 
 tances. The first of these is made evident from the following 
 considerations: Viewed from a given point, a series of equally 
 
24 A Mechanical Theory of the Solar Corona, 
 
 bright radial lines (and practically equidistant from the ob- 
 server) of indefinite extent will not appear equally bright to the 
 eye. The lines which are less inclined to the visual rays will 
 always be brighter than the more inclined ones, since in a given 
 direction the density of the luminous particles which make up 
 the line increases for decreasing inclinations. Hence, at a given 
 actual distance from the Sun, a streamer will appear brightest 
 when it is projected at the poles. 
 
 A second reason why the rays at the pole are the brightest is 
 the following: Beams from two sources of light of the same abso- 
 lute quantity, but of different angular magnitude, will be affected 
 differently in passing through an absorbing medium like our 
 atmosphere. If in one case the light is concentrated in a series 
 of points or lines in a given surface, and in the other case the 
 same amount of light is distributed over the whole of the same 
 surface, the absorbing effect of our atmosphere, through which 
 the two sets of luminous beams pass, will be more marked in the 
 latter case than it will in the former. Referring now to the 
 illustrations given in Plate VII., it will be noticed that the 
 number of rays increases with the inclination, while the mag- 
 nitude of these same rays decreases with the inclination. (In 
 order that the outer extremity of an inclined ray shall be at as 
 great a distance from the Sun's limb as a ray at the pole, the 
 ratio of the length of the former to the length of the latter 
 must be nearly as sec K to unity; or, for the same length of 
 inclined and polar rays, the corresponding ratio of distances 
 from the Sun's limb would be nearly as cos K to unity.) The 
 absorbing effect of our atmosphere will therefore always have 
 the tendency to cause a greater weakening of the more inclined 
 rays up to the position where the increase in density suddenly 
 rises in projection (in the model this absorptive effect is of 
 course not included, hence the gaps are not strongly shown), 
 the effect of which is to cause a great contrast between the 
 comparatively bright boundaries of the wings and the more 
 conspicuous polar rays. An inspection of the best photographs, 
 however, will show that these spaces, apparently devoid of 
 dense coronal matter, are often really filled out with faint rays, 
 which in a poor photograph would not show at all. The exist- 
 ence of rifts in other positions is illustrated in the models. 
 When the density of the streamers is so small that no definite 
 
By J. M. Schaeberle. 25 
 
 boundary of a wing is formed, the rays may be visible to a polar 
 distance nearly equal to 90 q>. Even if the streamers are not 
 of a gaseous character, their boundaries will not be sharply de- 
 fined, since the variation in latitude results in intersections at 
 very small angles; any attempt, therefore, to follow the course of 
 any particular one of these streamers, projected as it is against 
 many others, may be wholly in vain. 
 
 GRAPHICAL REPRESENTATION OP POLAR RAYS. 
 
 In the model (Plate VI.) the diameter of the needles com- 
 pared with the diameter of the ball is too great to admit of the 
 use of a sufficient number to produce the phenomenon of inclined 
 rays satisfactorily. 
 
 I have, however, been able to fulfill the conditions imposed 
 by a modification of the graphical method. In projection, the 
 streamers near the poles will appear to be tangent to a small 
 circle concentric with the Sun, instead of passing through the 
 Sun's center. Accordingly, instead of using slightly inclined 
 needles radiating from the surface of a sphere, I formed a series 
 of alternate bright and dark lines on an opaque film on the 
 surface of a piece of plate glass. These radial lines (slightly 
 curved) were all drawn tangent to the same small circle, so that 
 for a given distance from the center of this circle the inclination 
 to a normal was approximately the same as required by theory. 
 On this plate a second plate copied from the first was laid, in 
 both the direct and reversed positions. 
 
 Now the condition which must be fulfilled is this: The lines 
 of one plate must all be inclined in the same direction with 
 reference to the lines of the other plate, and this arrangement 
 must still produce rays which incline away from the lines rep- 
 resenting the secondary poles, if the theory is true. While I 
 had no doubt, from theoretical considerations, as to what the 
 pictorial result of the combination of the two gratings would 
 be, I had hardly hoped to obtain such satisfactory results on a 
 first trial. In Plate VII., Figures I., II., III., IV., VIII., X., XL, 
 XV., a and c illustrate some of the many, forms which can be 
 produced by streamers having nearly the same longitude, but 
 different latitudes, the plates for this purpose being superposed 
 without reversing. The remaining prints V., VI., VII., IX., 
 
26 A Mechanical Theory of the Solar Corona, 
 
 XL, XII., XIII., 6, d, and e, illustrate some of the forms pro- 
 duced when the streamers of the nearer hemisphere are projected 
 against those of the farther hemisphere, the plates being reversed 
 for this purpose. The shifting of the secondary coronal pole is 
 also represented (see Figures VIII. and IX., Plate VII.) ; one set 
 of lines being slightly raised or lowered to correspond to a change 
 in the position of the Sun's axis, which, in projection, is sup- 
 posed to be parallel to the edge of the paper. 
 
 The original prints were made on platinum paper, the two 
 superposed gratings representing the negative. These prints, 
 like those of Plate VI. , were then fastened to a large sheet and 
 photographed so as to obtain on a single page all the different 
 forms shown in Plate VII. 
 
 If a set of alternate bright and dark straight lines radiating 
 from the same point is placed upon another set of lines also 
 radiating from a point, no curves will be formed so long as the 
 centers of divergence are in the same line of sight; but the 
 instant one center is shifted with reference to the other, curves 
 quite similar to those given in Plate VII. will be produced. 
 If the plates are separated from each other, the errors in the 
 directions and positions of the lines will have a much less dis- 
 torting effect upon the rays than when the plates are close 
 together. 
 
 For the purpose of making the prints, however, it was nec- 
 essary to place the plates close to the platinum paper; all the 
 irregularities are therefore shown in these prints. It must be 
 remembered that from the nature of the model the outer por- 
 tions of the rays shown in the figures of Plate VII. are much 
 too bright and prominent, since the intensity does not diminish 
 with the distance from the center of divergence, as it evidently 
 does for the actual corona. 
 
 Then, again, in the model the contrast between the bright 
 and dark lines is abrupt and sharp, so that in the prints, for a 
 near view, these lines show plainly with the rays] but in the 
 case of actual streamers this state of things would evidently 
 not exist. If the plate is removed several feet from the eye, 
 only the rays will be distinguishable to the unaided vision. 
 
 NOTE. If each eruption is regarded as resulting in a large number of slightly 
 diverging pencils of matter, two eruptions on the central meridian in slightly 
 different latitudes would, in projection at the poles, also cause the phenome- 
 non of inclined rays. 
 
By /. M. Schaeberle. 27 
 
 If several plates are superposed, the contrasts are strength- 
 ened, and grotesque figures, resulting from irregularities in the 
 gratings, are not uncommon. r ,g 
 
 The boundaries of the wings in tne hemisphere which con- 
 tains the visible pole will, even for comparatively small values 
 of /?, usually appear double or multiple on the side turned 
 toward the adjacent pole, as portions of both sets of streamers 
 can be seen directly. 
 
 When fi is exactly zero, and the density of the streamers 
 uniform and great, the outer boundary of the wings, even at the 
 poles, may inclose the area usually occupied by the polar rays. 
 The density of the coronal areas will then increase toward those 
 axes whose latitudes are the same as the mean latitudes of the 
 streamers. The outlines of the fainter and more distant por- 
 tions, rendered visible by the overlapping of many faint streamers 
 projected along these axes, will then be approximately repre- 
 sented by equation (34). If, however, for any cause the stream- 
 ers from the low latitudes are not decidedly more conspicuous 
 than those in higher latitudes, or if there is a real or an apparent 
 periodic increase in the latitudes of the streams, which at times 
 transfers the scene of greatest visible activity to the higher limits 
 of the spot-zones, the outline of the bounding curves of the wings 
 is more likely to form a closed figure, and the general coronal 
 boundary will be more nearly circular, since the contrast 
 between the areas of different brightness for increasing dis- 
 tances may not be sufficient to bring out the envelope whose 
 approximate equation has already been considered. 
 
 From what has already been said, it is evident that the con- 
 dition of the Earth's atmosphere at the place and time of obser- 
 vation, determines, in a very great degree, the form of the visible 
 boundary of the outer corona. 
 
 Before giving further results of the comparison between 
 theory and observation, it is proper to make a few remarks 
 relating to the difficulty under which one labors in making an 
 unbiased examination of published data which are believed to 
 be representations of a certain theoretical form. 
 
 One is very apt to overestimate the importance of certain 
 coincidences and to disregard as unimportant certain discrep- 
 ancies. For instance, some of the evidence tending to prove 
 that the streamers issue from the Sun's equatorial regions 
 
28 A Mechanical Theory of the Solar Corona, 
 
 depends to a certain extent upon the observed variations of the 
 general outlines of the outer corona for different values of ft. 
 So far as this particular evidence is concerned the conclusion 
 reached must depend largely upon the data derived from mere 
 drawings, as the number of reliable photographs secured, for 
 different eclipses, is hardly sufficient to decide the case def- 
 initely. The interpretation of these drawings and poorer pho- 
 tographs, by different unbiased observers, will not necessarily 
 be the same. 
 
 Certain features should, however, always be present in good 
 photographs, giving evidence tending to prove that the stream- 
 ers originate in the spot-zones; of these the equatorial outlines 
 of the outer corona are perhaps the most important. The vari- 
 ation in the position of the secondary coronal poles, according 
 to the position of the observer with reference to the Sun's equa- 
 tor, also gives decisive evidence on this point. 
 
 For those eclipses of which we have photographs showing 
 polar rays suitable for measurement, I have used the constants 
 for a mean latitude of 15 for the streamers, and placed /? 
 for finding^. As the exact nature of the distribution of the 
 streamers in latitude is not known, any attempt to take into 
 consideration the slight variations in K, due to variations of ft 
 between and 8, in order to obtain a closer agreement between 
 theory and observation, would seem to be superfluous. For 
 those photographs of eclipses in which the rays are not shown, 
 I have used the following method for obtaining the direction 
 of inclination of the axis of the inner corona with reference to 
 the Sun's axis. At the origin of the curves bounding the wings 
 adjacent to the poles, tangent lines were drawn. Through the 
 two points of intersection, formed by the two lines near each 
 pole, the coronal axis is supposed to pass; when this line does 
 not pass through the Sun's center, I have assumed it to be par- 
 allel to the coronal axis, as the parallactic effect in a typical 
 form is such that both poles of the corona are shifted towards 
 the east of the points they would occupy in orthographic pro- 
 jection. The data for finding the position of the Sun's center 
 with reference to the center of the Moon is, also, nearly always 
 wanting. 
 
 As this shifting of the secondary poles will ordinarily be 
 only a few degrees, I have preferred to refer all the measures 
 
By J. M. Schaeberle. 
 
 29 
 
 to the true solar poles, rather than to the coronal poles; this 
 treatment, when fl is large, will of course make the differences 
 between computation and observation larger than they would 
 be if the coronal poles were taken as the origin. 
 
 For the above named conditions we therefore have for the 
 neighborhood of the poles and at the Sun's limb the following 
 numerical values: 
 tan l f 
 
 Hence, 
 
 -=c=1.5, a=1.2, and I l =2 
 
 tan ^1-5 sin d* p- 
 
 C08 (1.2 p) 
 
 o. 
 
 (36) 
 
 The following are the numerical values of K for each 10 of 
 distance from the adjacent pole: 
 
 P 
 
 10 
 
 20 
 
 30 
 
 40 
 
 K 
 
 15 
 
 31 
 
 46 
 
 58 
 
 With reference to the form of the outer corona, as shown by 
 any particular photograph, a very great deal depends upon the 
 mode of development of the photographic plate. A corona with 
 great equatorial extension can be made to appear nearly circu- 
 lar, especially if the atmospheric conditions at the time of an 
 eclipse are poor, so that only the brighter parts are shown. In 
 dealing with drawings not made from photographs, a rude 
 approximation to the true general outline can, at best, only be 
 obtained by combining the results of many different observers 
 of the same eclipse. Such drawings are of little or no value 
 for determining the sign of ft. The individual results of pure 
 drawings are often so discordant among themselves that it is 
 difficult to comprehend how they can possibly be reconciled so 
 as to represent the same phenomenon. 
 
 Bearing in mind the fact that in the graphical represen- 
 tation the intensity or distinctness of each streamer does not 
 decrease as the distance from the center increases, the prints 
 from the model, of which illustrations are given in Plate VI., 
 can, in a general way, be used to represent typical coronas. 
 The outline of the outer corona gives a rough indication of the 
 amount of the inclination of the Sun's axis; while the indi- 
 
30 A Mechanical Theory of the Solar Corona, 
 
 cated positions of the poles of the inner corona at once 
 determine whether the observer is above or below the plane of 
 the equator. It will be noticed that even in this simple model 
 the boundaries of the wings indicate the positions of the 
 coronal poles. The reason why the polar rays are not satis- 
 factorily formed has already been stated. 
 
 TABLE V. (Plate VI.) 
 
 No. of Figure. Corresponds to Season of the Year. 
 
 1-- 
 
 End of November and beginning of December. 
 
 I 
 
 End of May and beginning of June. 
 
 2, 3, and 4 
 
 End of December and beginning of January. 
 
 5 
 
 End of July and beginning of August. 
 
 6 
 
 End of January and beginning of February. 
 
 7 
 
 . _ . End of February and oeginning of March. 
 
 8 
 
 . - End of August and beginning of September 
 
 
 
 OBSERVATIONS OF SOLAR ECLIPSES AND COMPARISONS WITH 
 THEORY.* 
 
 Eclipse of 1715, May 2. 
 
 The individual drawings and descriptions are discordant 
 among themselves. 
 
 1776, February 9. 
 
 Same remarks as above. 
 
 1806, June 16. 
 
 The drawing is described as being a mere diagram, showing 
 the general effect of the light about the dark Moon. Equatorial 
 streamers evidently not markedly large. (Maximum of spots, 
 1804; minimum, 1811.) 
 
 1842, July 8. 
 
 Imperfect records. The observers supposed the equatorial 
 extension to be the zodiacal light. Equatorial extension great. 
 (Maximum of spots, 1837; minimum, 1843.) 
 
 * All the references are to A. C. RANYARD'S work " Observations made during 
 Solar Eclipses," Mem. R. A. S., Vol. XLL, unless otherwise stated. To make 
 possible a comparison between the variations in the general details of the 
 corona, as influenced by the phenomena of outgoing streamers retarded by 
 incoming ones (see postscript), I have added the times of maxima and minima 
 of Sun spots, as determined by the investigations of SCHWA BE, WOLF, and 
 others. 
 
By J. M. Schaeberle. 31 
 
 1851, July 28. 
 
 The daguerreotype taken at Konigsberg shows only the inner 
 and brighter portions of the corona. The resemblance to the 
 inner corona of July, 1878, is apparent. As required by the 
 theory for a typical form, when ft is positive, the axis of the 
 inner corona is inclined eastward, according to the inclination 
 of the faint boundaries of the wings just visible. (Maximum 
 of spots, 1848; minimum, 1856.) 
 
 1853, November 80. 
 
 RANYARD gives no drawing of this eclipse, but the description 
 is just what the theory requires it to be when the Earth is near 
 the node. (Five years after maximum, and three years before 
 minimum of spots.) 
 
 1858, September 7. 
 
 The drawing in general agrees with the form required by 
 theory 'when ft is large. (Compare with Figure 7, Plate I.) 
 (Minimum of spots, 1856; maximum, 1860.) 
 
 I860, July 18. 
 
 The photograph by MONSERRAT shows only the innermost 
 parts of the corona. The faint traces of the wing boundaries 
 indicate that the inner coronal axis is inclined towards the east, 
 as it should be. The drawings are very discordant among them- 
 selves, and show much polar extension. The majority of the 
 drawings also give an eastward inclination to the coronal axis. 
 Many of the drawings, especially the grotesque one by TEMPEL, 
 I can reproduce almost exactly, with a model* (Maximum of 
 spots, 1860.) 
 
 1867, August 18. 
 
 No photographs; the one drawing, which does not agree with 
 the description by the observer, shows great equatorial exten- 
 sion, with only a few short rays at the pole. (Minimum of 
 spots, 1867.5.) 
 
 1868, August 18. 
 
 All the drawings agree in giving evidence of a considerable 
 inclination of the Sun's axis. (Compare the drawings with the 
 denser portions of Figure 5, Plate I.) The majority of the 
 drawings give evidence of an eastward inclination of the inner 
 
32 A Mechanical Theory of the Solar Corona, 
 
 coronal axis corresponding to a positive value of ft. (Mini- 
 mum of spots, 1867.5; maximum, 1870.5.) 
 
 1869, August 7. 
 
 Both the photograph and the drawing are in accord with the 
 theory. In the drawing of MEEK and SCHOTT the extension in 
 the equatorial region is greatest at the equator, while the gen- 
 eral outlines, both in the photograph and drawing, are quite 
 similar to the inner portions of Figure 5, Plate I. According 
 to the boundaries of the wings as shown on WHIPPLE'S photo- 
 graph, the axis of the inner corona is inclined eastward as 
 required by theory when ft is positive. (Minimum of spots, 
 1867.5; maximum, 1870.5.) 
 
 1870, December 22. 
 
 The WILLARD photograph of this eclipse, aside from the fact 
 that the polar rays and other fine details of structure are wholly 
 wanting, is strikingly similar to our Cayenne photographs, when 
 they are printed so as to show only the same extent. No fine 
 coronal details are shown. On the BROTHERS photograph the 
 Sun seems to have reappeared during the exposure. The vari- 
 ous drawings of the outer corona are discordant among them- 
 selves. As required by theory, when ft is negative, the wing 
 boundaries indicate that the axis of the inner corona is inclined 
 towards the west. (Maximum of spots, 1870.5.) 
 
 1871, December 12. 
 
 The four excellent drawings by DAWSON, give a quadrilateral 
 outline to the outer corona, but the others do not as a rule give 
 evidence of a small value for ft. The photographs taken at 
 Baikul and Dodabetta, however, give almost exactly the same 
 detail. Practically the same structure is shown at both poles, 
 so that a superficial inspection of the photographs alone will 
 not determine which pole is turned towards the Earth. The 
 boundaries of the wings extend quite up to the poles, so that 
 only the nearly normal rays at the poles are shown on the pho- 
 tographs. At the time of this eclipse, the Earth was nearer to 
 the plane of the Sun's equator than it was at any previous or 
 subsequent eclipse of which photographs were obtained. Com- 
 pared with the equatorial extension, the polar extent was very 
 
By J. M. Schaeberle. 33 
 
 large. The observed general outline can, however, be fully sat- 
 isfied by supposing the eruptive forces to have been apparently 
 more active near the superior limits of the Sun-spot zone, or 
 that the equatorial streamers suffered retardation by coming 
 into contact with returning streams. As the original negatives 
 were only three tenths of an inch in diameter, considerable 
 uncertainty may exist in the fainter outlines of the enlarged 
 prints: I was so fortunate as to have the use of a good positive, 
 kindly loaned to me by Mrs. R. A. PROCTOR (evidently made 
 from a copy of the original negative and enlarged about three 
 diameters). I made a number of shorter timed negatives, and 
 these give, near the Sun, greater excess of equatorial extent 
 and less of polar extent than a positive of the January 1, 1889, 
 eclipse treated in the same way, but the boundaries are not as 
 sharply defined in the former case as they are in the latter. 
 From experiments with a model, the forms of the more lumi- 
 nous confused masses are so plainly the result of apparent inter- 
 sections of streamers (some of which are doubtless descending) 
 that a more detailed comparison can be dispensed with, since 
 no known forces could possibly account for the various forms 
 which can be traced on this photograph, if we suppose the 
 observed outlines to represent actual forms in space. As 
 required by the theory, the axis of the inner corona is nearly 
 coincident with the Sun's axis, being slightly inclined towards 
 the west, ft having a negative value. (Maximum of spots, 
 1870.5; minimum, 1879.0.) 
 
 187J, April 16. 
 
 There were no photographs taken at this eclipse. It is 
 described as being similar to the eclipse of August, 1868. As 
 drawn by STONE, the axis of the inner corona, given by RAN- 
 YARD, is inclined towards the west, in agreement with the typ- 
 ical form. A mere drawing, however, is deserving of but little 
 weight so far as the location of this axis is concerned.* 
 
 1875, April 6. 
 
 This eclipse is described as being curiously similar to the one 
 of the previous year (April 16). The photographs by LOCKYER 
 
 *NOTE. The observed data^for comparison between theory and observation, 
 up to and including this date, have been taken from RANYARD'S excellent 
 work on Solar Eclipses. 
 
34 A Mechanical Theory of the Solar Corona, 
 
 and SCHUSTER give unmistakable evidence of a considerable 
 inclination of the Sun's axis, as shown by the boundaries of the 
 wings. The telescopes were not driven by clockwork, so that 
 the polar rays are not shown. The direction of the Sun's axis 
 is not indicated with the necessary accuracy to determine the 
 inclination of the inner coronal axis with certainty. (Maxi- 
 mum spots, 1870.5; minimum, 1879.0.) 
 
 1878, July 29. 
 
 The first available photographs on which the finer details of 
 the structure at the poles are of such a character as to permit 
 of their being used to test the correctness of the theory were 
 obtained at this eclipse. In the drawing, made by Professor 
 HARKNESS, from the photographs obtained at this eclipse, pub- 
 lished in the volume issued by the United States Naval Observ- 
 atory, the polar streamers are so well defined that I have been 
 able to make the following comparison between the observed (o) 
 and computed (c) values of K. As the measures are necessa- 
 rily only approximate, I have recorded the measured inclina- 
 tions to the nearest five degrees for each ten degrees of polar 
 distance available: 
 
 1878, July 29 Observed and Computed Values of K. 
 
 DISTANCES FROM NEARER POLE. 
 
 QUADRANT. 
 
 
 10 
 
 
 
 20 
 
 
 
 30 
 
 
 40 
 
 
 
 
 
 c 
 
 C 
 
 
 
 c 
 
 o c 
 
 
 
 c 
 
 C 
 
 c 
 
 oc 
 
 I. . 
 
 10 
 
 15 
 
 5 
 
 30 
 
 31 
 
 1 
 
 40 
 
 46 
 
 6 45 
 
 58 
 
 13 
 
 II. 
 
 10 
 
 15 
 
 5 
 
 15 
 
 31 
 
 16 
 
 35 
 
 46 
 
 11 
 
 
 
 III 
 
 15 
 
 15 
 
 
 
 ?0 
 
 31 
 
 11 
 
 35 
 
 46 
 
 11 
 
 
 
 IV 
 
 10 
 
 15 
 
 5 
 
 
 
 
 
 
 "j 
 
 
 
 
 
 
 
 
 
 
 
 
 i ! 
 
 
 
 The observed values in the above table were taken from the 
 composite drawing (Plate XX., Fig. 3), found in United States 
 Naval Observatory publication on the " Solar Eclipse of July 
 29, 1878." The photographs taken at different stations are in 
 accord with each other. The numerous drawings are, as usual, 
 very discordant among themselves, but on the whole indicate 
 a comparatively large value of ft. The northern wings have 
 
By J. M. Schaeberle. 35 
 
 multiple boundaries on the sides towards the nearer pole. Both 
 coronal poles are east of the Sun's poles; but the line joining 
 the poles of the corona, as found by means of the boundaries of 
 the wings adjacent to the poles, is inclined towards the east, as 
 required for a typical form. The polar rays, alone considered, 
 w T ould indicate that the coronal axis was nearly parallel to the 
 Sun's axis. Professor HARKNESS calls attention to the fact that 
 the orientation may be several degrees in error, as only the 
 sides of the photographic plates could be used as guides. 
 
 The excellent drawing by TROUVELOT is in remarkable agree- 
 ment with the photographs. Naked-eye drawings by Professor 
 NEWCOMB and Professor LANGLEY, the latter being stationed on 
 the top of Pike's Peak, show the greatest equatorial extent of 
 the corona that has been observed up to the present time. 
 (Compare the drawing made from the photographs with Figure 
 5, Plate VI.) (Minimum of spots, 1879.0.) 
 
 1880, January 11. 
 
 The sketches made during this eclipse show how untrustwor- 
 thy the data derived from mere drawings can be. In Professor 
 DAVIDSON'S Report (see United States Coast and Geodetic Sur- 
 vey Report for 1882), the corona is outlined as in Figure 2 or 
 3, Plate VI., while in the sketches of Lieutenant CHRISTOPHER 
 only a narrow, nearly concentric band of light is shown sur- 
 rounding the dark limb of the Moon. There are no photo- 
 graphs of this eclipse, so far as known. 
 
 1882, May 17. 
 
 The drawing, made from photographs, given in Captain AB- 
 NEY'S and Dr. SCHUSTER'S report on the solar eclipse, is exceed- 
 ingly interesting as representing an arrangement of streamers 
 similar to the conditions found in the model; that is, no polar 
 rays are formed. The general appearance of the nearly circular 
 outline of the corona would seem to indicate that the solar 
 activity was considerable in the higher limits of the spot-zones; 
 or, what is more probable, that those in the middle latitudes 
 were retarded, and that the incoming streamers were more 
 numerous at the equator (see Postscript), and that in both 
 zones only the more conspicuous streamers of the farther hemi- 
 sphere are seen in projection through the dense masses of 
 
 4 
 
36 A Mechanical Theory of the Solar Corona, 
 
 rays, most of which are apparently wholly formed in the nearer 
 hemisphere. There is strong evidence tending to show that 
 there were numerous collisions, as in many places the corona 
 is brightest at some distance from the Moon's outline. I can 
 almost exactly reproduce this coronal form with a model. The 
 inner portions of Figure 2, Plate VI., are quite suggestive. 
 (Minimum of spots, 1879; maximum, 1883.) 
 
 1888, May 6. 
 
 The drawings made by Professor TACCHINI and Dr. DIXON give 
 evidence of a considerable inclination of the Sun's axis. The 
 Lick Observatory does not possess any photographs. 
 
 Note October, 1890. Since the above was written the 1889 
 volume of the " Philosophical Transactions" has been received. 
 Like the previous eclipse (1882) there is much confusion in the 
 coronal detail. If the orientation of the enlarged drawing made 
 from photographs secured by the English party is correct as 
 published in this volume, then we have here a case of a nearly 
 circular corona, in which the less inclined streamers projected 
 near the north pole of the Sun had a decidedly greater density 
 than the more inclined ones; since the apparent north coronal 
 pole is at the enormous distance of nearly 20 to the east of the 
 Sun's pole!! At the south pole the structure is more nearly 
 radial, so that the position of the south coronal pole can not be 
 located with great accuracy. 
 
 Note No. 2 March, 1891. Since the above has been in type 
 I have, very fortunately, found a photographic print of this 
 eclipse (see Annuaire, Bureau des Longitudes, 1884), made by 
 M. JANSSEN from a negative which he exposed during the whole 
 time of totality. Although the orientation can only be made 
 by means of the edges of the card on which the print is mounted 
 (the north pole is toward the lower left-hand corner of the 
 card), still all doubt as to the proper orientation of the above 
 mentioned drawings seems to be removed. Dr. DIXON observed 
 with a diagonal eye-piece; the top of his drawing should there- 
 fore be marked S. instead of N. 
 
 In Mr. WESLEY'S drawing (Phil. Trans.) the top of the page 
 is N. The axis is correctly orientated. 
 
 The remarkable difference which can exist between the draw- 
 yng of a photograph taken with one instrument and a photo- 
 graphic print from a negative taken with another instrument is 
 
By J. M. Schaeberle. 
 
 37 
 
 well illustrated in this case. In JANSSEN'S print not only are 
 the symmetrical wings and fish tails shown, but even the trumpet- 
 shaped outlines, on the east side at least, are unmistakable. The 
 extended luminosity near the north pole would seem to indicate 
 that the streamers were not uniformly distributed in longitude, 
 so that the ratio of D to D' is no longer that which holds good for a 
 typical corona. Now, comparing Mr. WESLEY'S drawing with 
 the print, an apparent tendency to form two north and, less 
 conspicuously, two south coronal poles adjacent to the bound- 
 aries of the wings is at once recognized. The reversal of the 
 inclinations near the north pole is very marked and is due to 
 an irregular variation in the density of the streamers, as can be 
 readily shown with the aid of the models (see, for instance, 
 Plate VII., Fig. a). The tendency to form multiple poles is 
 also partially shown in some of the figures of Plate VI. 
 
 The axis, which is symmetrically situated with reference to 
 these poles, in Mr. WESLEY'S drawing, is inclined towards the 
 west, in agreement with the theory. Compare the print with 
 Figures 3, 4, and 6 of Plate VI., disregarding the unsymmetrical 
 luminosity above the north pole. (Maximum of spots, 1883.) 
 
 1886, August 29. 
 
 The photographs taken by PICKERING on the Island of Gre- 
 nada show enough of the finer detail in certain places near the 
 poles to allow of a rough comparison between theory and obser- 
 vation. The inclinations (to the nearest, 5) of the polar rays 
 to normals through their bases are given in the table below for 
 each 10 of polar distance applicable. The measures are made 
 from the drawing given in the frontispiece of Harvard College 
 Observatory Annals, Vol. XVIII., No. V. 
 
 DISTANCE FROM NEARER POLE. 
 
 QUADRANT. 
 
 
 10 
 
 
 
 20 
 
 
 
 30 
 
 
 
 40 
 
 
 
 o 
 
 c 
 
 C 
 
 
 
 c 
 
 o c 
 
 
 
 c 
 
 o c 
 
 
 
 c 
 
 o c 
 
 I. . 
 
 15 
 
 15 
 
 
 
 25 
 
 31 
 
 6 
 
 
 
 
 
 
 
 II. 
 
 15 
 
 15 
 
 
 
 20 
 
 31 
 
 11 
 
 50 
 
 46 
 
 4-4 
 
 
 
 
 Ill; 
 
 10 
 
 15 
 
 5 
 
 
 
 
 
 
 
 
 
 
 IV. 
 
 ? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
38 A Mechanical Theory of the Solar Corona, 
 
 In the second quadrant, at 30 polar distance, the rays are 
 not distinct, being nearly lost in the general illumination; the 
 measure is therefore unreliable. Just at the south pole there is 
 a cone with the sides inclined 10 to the normal, apparently 
 made by the overlapping of rays. On the west side of the north 
 pole the distribution of the streamers is apparently too irregu- 
 lar to produce even a trace of the symmetrical inclined rays; 
 this supposition is further strengthened by the fact that at 25 
 polar distance there is a great wing extending to a distance 
 nearly equal to the equatorial extent of the corona. As required 
 by the theory when fi is positive, the axis of the inner corona 
 is inclined towards the east. (Maximum of spots, 1883; mini- 
 mum, 1888 or 1889.) 
 
 Note October, 1890. Since the above was written the volume 
 of the PhiL Trans., which contains a drawing of a photograph 
 of this corona (discussed by Messrs. DARWIN, SCHUSTER, and 
 MAUNDER), has appeared. The orientation differs something 
 less than 4 from the PICKERING photograph, so that for this 
 copy of the corona the coronal axis is slightly inclined west- 
 ward. The following measures can be compared with those in 
 the preceding table: 
 
 QUADRANT. 
 
 DISTANCE FROM NEARER POLE. 
 
 10 
 
 20 
 
 30 
 
 40 
 
 
 
 c 
 
 o c 
 
 
 
 c 
 
 C 
 
 
 
 c 
 
 o c 
 
 
 
 c 
 
 C 
 
 I. . 
 
 20 
 20 
 10 
 
 
 15 
 15 
 15 
 15 
 
 4-5 
 
 ! 
 
 15 
 
 40 
 30 
 20 
 20 
 
 31 
 31 
 31 
 31 
 
 4- 9 
 1 
 11 
 
 
 
 
 
 
 
 II 
 
 50 
 
 46 
 
 +4 
 
 
 
 
 III. .. 
 
 
 
 
 IV 
 
 11 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 If the orientation is changed so as to agree with PICKERING'S 
 diagram, the above residuals will be reduced in magnitude some- 
 what. 
 
 1887, August 18-19. 
 
 The best photographs taken at Jourgewetz by Dr. BELOPOLSKY 
 are similar to those of the previous year. A certain con- 
 fusion is apparent, so that no great amount of detail is shown 
 at the poles. The. axis of the inner corona apparently had a 
 
By J. M. Schaeberle. 
 
 39 
 
 decided inclination toward the east, as required by the theory 
 when ft is positive. Dr. GLASENAPP'S results give the same evi- 
 dence. (Minimum of Sun spots, probably 1888 or 1889.) 
 
 1889) January 1. 
 
 It is no exaggeration to say that the photographs taken dur- 
 ing this eclipse are better than any which had previously been 
 taken. On the magnificent negatives secured by BARNARD, 
 BURCKHALTER, CHARROPpiN, Harvard College Observatory 
 party, and others, the coronal structures are shown in a way 
 that leaves but little to be desired. 
 
 In the following table the comparison between theory and 
 observation will prove interesting: 
 
 QUADRANT. 
 
 DISTANCE FROM NEARER POLE. 
 
 10 
 
 20 
 
 30 
 
 40 
 
 
 
 c 
 
 o c 
 
 
 
 c 
 
 C 
 
 
 
 c 
 
 C 
 
 o 
 
 c 
 
 o c 
 
 I 
 
 20 
 20 
 10 
 10 
 
 15 
 15 
 15 
 15 
 
 -j-5 
 
 s 
 
 5 
 
 30 
 
 31 
 
 1 
 
 35 
 
 46 11 
 
 50 
 
 58 
 
 8 
 
 II. 
 
 
 III. 
 
 25 
 25 
 
 31 
 31 
 
 6 
 6 
 
 
 
 
 
 
 
 IV. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I have only used those polar distances at which the indi- 
 vidual rays could still be recognized. The corona was nearly 
 of the typical form. The great equatorial extension with the 
 trumpet-shaped outlines has just the form which streamers 
 of indefinite length from low latitudes will produce in pro- 
 jection. With the increase of distance the illumination 
 gradually fades out, but towards the axes of greatest den- 
 sity in projection the increasing number of overlapping 
 streamers causes a gradual increase in the illumination, result- 
 ing in the trumpet-shaped outlines which form so promi- 
 nent a feature of this eclipse. For all coronas, in which the 
 fish tails are prominent, this particular feature should here- 
 after be shown in cases where the observers are favored with 
 exceptionally good atmospheric conditions. This was the case 
 for the present eclipse; the clear California sky revealed out- 
 lines not before seen on photographs taken during sun-spot 
 minima. 
 
40 A Mechanical Theory of the Solar Corona, 
 
 Not only is this phenomenon a necessary consequence of the 
 theory, but all doubt as to its nature would seem to be removed 
 by the appearance of the photographs themselves; numerous 
 streamers, of such a length and form as to preclude the idea 
 that they coincide with magnetic lines of force, are seen radiat- 
 ing from parts of the Sun's limb, which indicate that the 
 origins are in the low latitudes. Here, again, the model need 
 only be compared with the photographs, to test the agreement 
 of the theory with observation. The boundaries of the wings 
 in the southern hemisphere are multiple on the sides turned 
 toward the nearer pole. The axis of the inner corona, as 
 required by the typical form, is inclined toward the west, the 
 Earth being below the Sun's equator. (Minimum of spots, 
 1888 or 1889.) 
 
 1889, December 21, 22. 
 
 The photographs taken at Cayenne again show the structure 
 of the corona with such minuteness of detail that a direct com- 
 parison between the observed and theoretical inclinations of the 
 polar rays can also be used to test the theory. 
 
 As the position of the meridian is not indicated on the prints, 
 the photographs can be accurately orientated by means of the 
 following datum. The center of the heaviest north polar ray 
 (the most westerly one of the more conspicuous and nearly 
 normal rays) has a position angle of 4. 8; the pole of the Sim 
 is therefore 2 east of the heavy north polar ray. 
 
 DISTANCE FROM NEARER POLE. 
 
 QUADRANT. 
 
 
 10 
 
 
 
 20 
 
 
 
 30 
 
 
 
 40 
 
 
 
 
 
 c 
 
 o c 
 
 
 
 
 oc 
 
 
 
 c 
 
 C 
 
 
 
 c 
 
 o c 
 
 I. 
 
 10 
 
 15 
 
 5 
 
 20 
 
 31 
 
 11 
 
 40 
 
 46 
 
 6 
 
 
 
 
 11. ... 
 
 10 
 
 16 
 
 5 
 
 
 
 
 
 
 
 
 
 
 III. .. 
 
 9,0 
 
 15 
 
 4-5 
 
 25 
 
 31 
 
 6 
 
 45 
 
 46 
 
 i 
 
 
 
 
 IV 
 
 20 
 
 15 
 
 +5 
 
 35 
 
 81 
 
 -h 4 
 
 45 
 
 46 
 
 _1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Although the sky was apparently free from clouds during the 
 total phase of the eclipse, the air was so charged with moisture 
 that the atmospheric glare must have been very prominent. 
 Yet, notwithstanding these conditions, the equatorial extent of 
 the corona, as shown on the photographs, is more than a degree; 
 
By J. M. Schaeberle. 41 
 
 the wings are very conspicuous, and the polar rays are shown 
 with all desired distinctness. The trumpet- shaped outlines, so 
 prominent a feature in the previous eclipse, are lost in the sky 
 illumination, although faint traces of this phenomenon can be 
 distinguished. 
 
 In the southern zone there was evidently a vigorous outburst 
 near longitude r r =0. The protuberances (shown on the 
 negatives) projected at this part of the Moon's outline give 
 evidence to the same effect. The boundaries of the wings in 
 the southern hemisphere are multiple on the sides turned 
 towards the nearer pole. 
 
 As required by the theory, the axis of the inner corona is 
 inclined slightly towards the west, the Earth being below the 
 Sun's equator. (Minimum of spots, 1888 or 1889.) 
 
 Probable Form of the Solar Coronas of 1892 and 1893. 
 
 The next total eclipse of the Sun takes place in 1892, April 
 26. If observed, the following prediction as to its form should, 
 in general, be verified, under favorable atmospheric conditions: 
 Polar extent large. Inner coronal axis (if polar rays or 
 wings are shown) inclined westward. Rifts numerous. Indi- 
 cations of collisions. General form similar to Figure 7, Plate 
 VI. The same description will apply to the eclipse of April 
 15-16, 1893. 
 
 It will be noticed that I have used the same constants for 
 finding the value of Jc for a given p in all cases (see page 75). 
 A closer agreement between theory and observation could doubt- 
 less have been obtained by' varying the values of c and a (for 
 different values of /?) and taking into account the shifting of the 
 coronal pole; but as it appeared that the measured inclinations 
 were liable to errors, which as a rule much exceeded those due to 
 neglecting the terms which involved or caused the shifting of 
 the coronal poles, I deemed it best to refer all measures to the 
 Sun's true poles. The magnitude of the residuals obtained can 
 then be used to detect any abnormal discrepancies which may 
 result from some error in the orientation of the photograph. 
 
 The object of the present investigation was not to force an 
 exact agreement by the arbitrary variation of certain quanti- 
 ties, but to secure confirmatory or non-confirmatory evidence, 
 
42 A Mechanical Theory of the Solar Corona, 
 
 by comparing a purely theoretical investigation of a general 
 character with the results obtained from actual observation. 
 
 It is hardly necessary to state that nothing in this theory 
 requires that all the streamers shall be confined between any 
 fixed circles of latitude, nor that the apparent density and 
 magnitude of the streamers may not be subject to periodic 
 variations, due to physical causes which relate to the molecular 
 nature of the Sun's internal constitution, with which this in- 
 vestigation has nothing to do. (See Postscript.) 
 
 There seems to be no feature of the coronal structure which 
 cannot be accounted for in a satisfactory manner by this new theory 
 of the solar corona. 
 
 In conclusion, it gives me pleasure to thank Messrs. BURN- 
 HAM and BARNARD for their aid in the work of the photographic 
 reproductions, and Mr. KEELER for various information relating 
 to spectroscopic solar physics. 
 
 I am much indebted to Professor HOLDEN, who, during the 
 preparation of this memoir, has ever been ready to aid me in 
 various ways. 
 
 MT. HAMILTON, June, 1890. 
 
 J. M. SCHAEBERLE. 
 
 POSTSCRIPT. 
 
 I have spent some time in an extended examination of the 
 magnificent spectroscopic observations made at Rome, with a 
 view to determine, if possible, whether the solar prominences 
 observed in high latitudes could not be reconciled to the hypoth- 
 esis that they were but the projections of streams, which, issu- 
 ing from a low latitude, appeared in projection to come from 
 the limb of the Sun. In many instances this seems to be the 
 case, but the complications, especially in the low latitudes, 
 resulting from the perspective overlapping of these promi- 
 nences in largely different longitudes, is so great that much 
 uncertainty exists as to the motion of any given prominence. 
 The uncertainty could be settled at once by following a con- 
 spicuous polar prominence for several hours, and carefully 
 recording its position-angle from time to time. If the base of 
 such a prominence is in a low latitude, the position-angle 
 
By J. M. Schaeberle. 43 
 
 should change at the rate of about 2 an hour. I have not 
 had an opportunity to make an observation of this kind since 
 the theoretical investigation was completed. 
 
 The prominences at the poles are so rare, and the observa- 
 tions usually so short, that no definite conclusions can be drawn 
 from the data at hand. Again, prominences at the poles should, 
 on this hypothesis, always have large component velocities either 
 towards or away from the Earth. 
 
 During the total eclipse of August 29, 1886, Professor SCHUS- 
 TER observed the spectrum of a great prominence near the pole 
 of the Sun. His observations indicated a motion of two hun- 
 dred and forty-seven miles per second towards the observer. 
 Assuming the origin of the prominence to be in the spot-zone, 
 this velocity indicates roughly an initial parabolic velocity. 
 
 As bearing upon this point, th'e following extracts from 
 " Chemistry of the Sun," by that eminent solar physicist, J. 
 NORMAN LOCKYER, will be found very suggestive. The italics 
 are my own: . 
 
 Such prominences have been seen to mount upwards at the rate of 250 
 miles a second ; that is, nearly 1,000,000 miles an hour. * * * There 
 are indications that these prominences, instead of rising vertically, as we may 
 imagine them to do, are at times shot out sideways almost tangentially. In that 
 case, of course, the spectroscope enables us to determine the velocity. One 
 hundred miles a second, either towards or from the eye, is by no means an uncommon 
 velocity. * * * The height of some of these prominences is very great. 
 Professor YoiTNG records one seen in 1878 as being nearly 400,000 miles high ; 
 that is, 13^ minutes of arc, the solar radius being 16 minutes. 
 
 The actual velocity of motion will, in general, always be 
 greater than the observed, since the spectroscope only gives 
 that component which is parallel to the Earth's radius-vector, 
 while the direct visual observation only gives the component 
 at right angles to the same line. 
 
 The small inclined saw-tooth protuberances, as well as many 
 of the grotesque larger ones, are probably formed in much the 
 same way that the polar rays are produced; the sudden changes 
 of inclination being due to variations in the density of the 
 streams. It would seem to require a continuous and more or 
 less uniform series of eruptions in certain zones on the solar 
 surface to satisfactorily account for these ever-present smaller 
 protuberances, and the theory certainly demands such a dis- 
 tribution of the streamers. Let us now consider 
 
44 A Mechanical Theory of the Solar Corona, 
 
 SOME PHYSICAL PHENOMENA INVOLVED IN THE MECHANICAL 
 THEORY OF THE CORONA. 
 
 While I invite the severest criticism, from astronomers, of the 
 theory given in the preceding pages, I ask their indulgence with 
 reference to certain deductions which I have made, and now, 
 with great deference, submit to their attention. Assuming my 
 premises (The Mechanical Theory) to be true, I trust that in 
 deducing some of the resulting phenomena my arguments will 
 at least be found to have a logical sequence. 
 
 All the results obtained in the foregoing investigation have 
 been deduced on the hypothesis that the force of ejection is 
 such as to give a parabolic velocity to the streams; but so far 
 as the form near the Sun is concerned, an inspection of Table I. 
 at once shows that practically the same results would be obtained 
 on the hypothesis that the force is only just sufficient to give an 
 elliptical velocity corresponding to a period of only a few hours. 
 
 The moment the returning streams are taken into consider- 
 ation the effect is such as to cause a periodic variation in the 
 detail of the corona, and simultaneously to cause a periodic vari- 
 ation in the surface features of the Sun, as I shall now proceed 
 to show. 
 
 To whatever cause the eruptions in certain zones on the 
 Sun may be due, we are evidently justified in assuming that, 
 in the long run, the forces there at work have, at a given period 
 of the Sun's age, a mean value (F). One measure of this force 
 (F) will be the maximum distance (d) from the Sun's center 
 (=2a nearly), to which a given particle (mass=m) is projected 
 in the time \t (nearly), so that the periodic time will be t. As 
 has already been shown, the heliocentric latitude of an ejected 
 particle during its whole motion will remain nearly the same, 
 so that the latitudes of different parts of the same stream will 
 be nearly the same as the latitude of the point of ejection. Now, 
 if each stream of particles in a given zone is ejected by an 
 instantaneous force (F), the more advanced portions of the 
 streams will be unimpeded during the first half of their path ; 
 on returning, however, the chance of collision with the same or 
 other outgoing streams varies inversely as the square of the 
 distance from the Sun. 
 
 Near the Sun, therefore, collisions must occur which tend to 
 
By J. M. Schaeberle. 45 
 
 retard or stop the outgoing streams, resulting in a temporary 
 increase in the heat of the combined colliding masses (causing 
 a consequent increase in the brightness of the corona at such 
 places, and at the same time rendering the coronal detail more 
 confused). This heat will tend to be largely dissipated before 
 such masses fall back into the Sun, which they will then reach 
 with comparatively small velocity and low temperature. Unre- 
 tarded returning streams on striking the Sun will tend to greatly 
 raise the temperature at the points of impact; perturbed return- 
 ing streams could, of course, strike all parts of the Sun's surface, 
 but the general tendency of these perturbations will be to dimin- 
 ish the latitudes of the returning streams. Unperturbed return- 
 ing streams will always fall within the limits of the Sun-spot 
 zones. 
 
 So long as the incoming streams are very numerous, the out- 
 going ones will, in a great measure, be stopped, so that, after 
 the interval t, there will be comparatively few returning streams; 
 a direct consequence of this state of things is to allow free pas- 
 sage for the outgoing streams, which, since there are now but 
 few collisions, results in (1) an apparent diminution in the 
 brightness of the corona, (2) more regular and sharply defined 
 detail, and (3), in general, a more uniformly illuminated solar 
 surface (i. e., fewer solar spots). 
 
 The periodic character of this phenomenon can be well illus- 
 trated by means of a vertical jet of water. When the water is 
 first turned on it almost instantly shoots up to its maximum 
 height; the returning drops then gradually check the velocity 
 of the stream near the origin, so that after a moment the whole 
 mass seems to be piled up just a little above the orifice; before 
 the last portions of the still falling stream reach the retarded 
 stream the jet begins to resume its former activity, and then 
 again rises to a considerable height. These oscillations are 
 repeated at nearly uniform intervals, the period of one complete 
 phase being roughly equal to twice the time required for a given 
 drop to describe the whole path. 
 
 If the ejective force is such as to make t about five years, a 
 complete cycle of changes will take place in the time 2t, and 
 after the same manner as is observed in the Sun-spot cycle. It 
 is rather remarkable that the aphelion distance of the streams 
 corresponding to this value of t is nearly the same as Jupiter's 
 
46 A Mechanical Theory of the Solar Corona, 
 
 distance from the Sun; so that the perturbations produced by 
 this planet may have more to do with the regularity of the 
 period than the assumed constant force of ejection. The initial 
 velocity required to just carry a particle from the Sun to Jupiter 
 is but little less than a parabolic velocity. For an initial para- 
 bolic velocity Saturn, alone considered, would, on the same 
 hypothesis, cause a complete cycle of less marked changes in 
 twenty years, Uranus in sixty years, and Neptune in one hun- 
 dred and twenty years. The comparatively insignificant plan- 
 ets inside of the orbit of Jupiter would cause minor variations, 
 corresponding to cycles, which, even for Mars, would be of less 
 than two years' duration. 
 
 A maximum of Sun spots will therefore correspond to the 
 times when the returning streams are most numerous; the 
 corona at these same times will be brightest and most con- 
 fused near the Sun, and on account of the retardations, the 
 equatorial extent will, as a rule, be least. But at the equator 
 there will be more returning than outgoing streams (on account 
 of the planetary perturbations) ; consequently, the illumination 
 in this plane will be greater at this time than it is when only 
 outgoing streams are present. 
 
 A minimum of Sun spots will correspond to the times when 
 the incoming streams have been exhausted. Consequently, 
 the unimpeded outgoing streams will have their normal veloc- 
 ity and extent, the corona will show great equatorial extension, 
 and, as a rule, be more sharply defined and have more promi- 
 nent fish-tail outlines, especially when the Earth is near the 
 Sun's equator. 
 
 In the equation: 
 
 V increases as M diminishes. As it is probable that at the 
 instant of ejection the masses are of a gaseous character, the 
 velocity may be very great without necessarily requiring an 
 improbable ejective force. It may be well to call attention to 
 the fact that the initial velocity required to send a particle to a 
 distance of only one solar diameter from the Sun's surface is 
 already greater than four fifths of the velocity required to send 
 the same particle to an infinite distance. For lines of motion 
 
By J. M. Schaeberle. 47 
 
 which are not normal to the Sun's surface, much greater initial 
 velocities will be required to carry the particles to the same 
 distance. 
 
 On this hypothesis the duration of the Sun-spot period there- 
 fore indicates that the forces of ejection are such as to give the 
 streams a mean velocity but little less than that in a parabolic 
 orbit. Streamers having inclined initial directions of motion 
 will, as a rule, either be destroyed by others, or be so scattered 
 that no marked effects will be produced by them. (The zone 
 of maximum Sun spots is the same as the zone above which the 
 maximum number of collisions takes place.) As the Earth's 
 maximum distance from the plane of the Sun's equator is less 
 than 8, and as the zone of maximum activity on the Sun has 
 twice this latitude, the chance of the Earth passing through 
 an outgoing stream is less than it is for an incoming (perturbed) 
 stream. Such encounters must, however, take place. 
 
 THE ZODIACAL LIGHT AND THE GEGENSCHEIN. 
 
 According to the Mechanical Theory, the ejected streams of 
 matter, although they are of double curvature, will always be 
 directed nearly towards the Sun (except at aphelion). Let us 
 now consider some phenomena which can be produced experi- 
 mentally. The diffused light caused by a series of nearly 
 parallel luminous lines (of indefinite extent) very distant from 
 the observer will always be most conspicuous in those direc- 
 tions in which the lines are projected as mere points. If the 
 observer is placed within the space through which these lines 
 pass, there will be two points of equal maxima 180 apart; 
 but if the actual intrinsic brilliancy of the lines increases 
 from one extremity of the set to the other, then, of course, the 
 two maxima will differ enormously in brilliancy. If the depth 
 of these lines in cross-section is the same on all sides of the 
 observer, the illumination will be symmetrically arranged in 
 concentric zones, which gradually decrease in brightness as the 
 angular distance from their common pole (which corresponds 
 to the brighter maximum) increases, until the angular distance 
 is reached at which the apparent increase in brilliancy due to 
 projection more than compensates for the decrease in brilliancy 
 due to distance from the origin. If one diameter of the cross- 
 section at the observer is greater than any other, the excess in 
 depth produces an apparent increase of illumination in a plane 
 
48 A Mechanical Theory of the Solar Corona, 
 
 which contains this diameter and the vanishing points of the 
 lines, the law of variation of brightness in this plane being the 
 same as for a circular cross-section. If the observer is near, 
 but not within, the space traversed by the lines of indefinite 
 extent, no conspicuous variations will be apparent, although 
 one hemisphere will actually be brighter than the other. If 
 in each of the above cases the luminous lines be assumed to 
 radiate from a common center, which is at a great distance from 
 the point of observation, the distribution of those lines which 
 pass nearest to the observer will, on account of their small 
 distance, almost wholly determine the form of the resulting 
 phenomenon. 
 
 Let us now consider the phenomenon caused by the outgoing 
 and incoming streamers from the Sun. In December and June 
 the Earth is in the plane of the Sun's equator and at its maxi- 
 mum distance from the nearest zone of streamers, and it is 
 symmetrically situated with reference to both zones. Now, 
 since the Earth is in neither of these zones, no great contrasts 
 of light will be produced on the side opposite to the Sun. Even 
 if there were no streamers near the Earth, the light in the 
 two hemispheres would be so diffused (covering the whole visible 
 sky with the exception of an ill-defined less luminous band 
 along the ecliptic) that it is quite probable the variations in 
 light would not be apparent, except at the time of an eclipse of 
 the Sun, when typical coronal forms would be seen. But the 
 arrangement of the comparatively few streamers which pass 
 near the Earth on all sides will, on account of their nearness, 
 now determine the resulting form of the outlines. The density 
 in projection will evidently be greatest in the plane of the Sun's 
 equator, but this density will be at a minimum. 
 
 When the Earth is at its greatest distance above (September) 
 or below (March) the plane of the Sun's equator, it will be at 
 its least distance from the zone of maximum density of the 
 streamers. In projection, the density will now be greatest in 
 a plane which is practically coincident with the plane of the 
 ecliptic; the excess of streamers in the direction of maximum 
 density will, however, have a tendency to cause the center of 
 illumination to be shifted slightly above the plane of the ecliptic 
 in March and slightly below this plane in September. (The effect 
 of the decrease in latitude of the returning streams will have a 
 precisely similar tendency.) In general, at any time of the year, 
 
By J. M. Schaeberle. 
 
 49 
 
 the illumination at any angular distance from the Sun will be 
 greatest near a plane which contains the radius-vector of the Earth, 
 and which is perpendicular to the orthographic projection of the 
 Sun's axis. 
 
 The phenomenon will be least conspicuous in December and 
 June, and most conspicuous in September and March. It will 
 always be brightest near the Sun, and gradually diminish in 
 intensity to within a certain distance of the point which is 
 180 from the Sun, where theoretically there will be a slight 
 increase in the luminosity. These conditions are in agreement 
 with actual observation. 
 
 Now, the effect of the perturbations of the superior planets is 
 always such that the latitude of a normally ejected returning 
 stream will be less than that of the same outgoing stream; con- 
 sequently, these streams will be more conspicuous to an observer 
 in a smaller heliocentric latitude than the outgoing streams. If 
 lines are drawn from the Earth tangent to the incoming stream, 
 these lines will.aM have a slightly less longitude than that of 
 the Earth, so that the center of the secondary maximum will 
 always be west of the point which is 180 from the Sun, if the 
 returning stream is more conspicuous than the outgoing one, 
 since lines from the Earth drawn tangent to the outgoing stream 
 will always lie east of the same point. The greatest extent will 
 be in longitude. 
 
 To show how this theory agrees with observation, I give be- 
 low a resume of the published results of the most noted 
 observers of the Gegenschein. I have taken the mean results 
 for two series, one corresponding to the months when the Earth 
 was above the Sun's equator, the other when it was below this 
 plane: 
 
 
 ow2 
 
 si 
 
 ^ 
 
 g 
 
 
 
 Teriod. 
 
 pa 
 
 || 
 
 If 
 11 
 
 2, 
 
 Observers. 
 
 Reference. 
 
 
 ||| 
 
 2.2, 
 
 CD 
 
 s 
 
 
 
 Mar. to April 
 
 +1-.9 
 
 -1.5 
 
 
 
 12 
 
 SCHMIDT _. 
 
 .. Astr. 
 
 Aug. to ISov. 
 Feb. to May. 
 
 +2.1 
 0.0 
 
 +0 .2 
 2.0 
 
 i 
 
 33 
 32 
 
 SEARLE & 
 
 Nach.,No.l726. 
 
 Aug. to Oct.. 
 
 0.2 
 
 +0.2 
 
 + 
 
 11 
 
 I" WENDELL.. 
 
 ...Astr. Nach., 
 Nos. 2376-2441. 
 
 Sept. to Oct. . 
 
 +2.8 
 
 0.7 
 
 
 
 4 
 
 [ BARNARD . 
 
 Astr. Jour- 
 
 Feb. to Mar. 
 
 +0 .1 
 
 +0.8 
 
 + 
 
 11 
 
 
 nal, No. 168. 
 
50 A Mechanical Theory of the Solar Corona, 
 
 While the individual observations on any given day are 
 often very discordant, the above mean results certainly show a 
 remarkable agreement with the theory. 
 
 When one considers that the area covered by the Gegenschein 
 is often as much as 20 long by 10 or 15 wide, the difficulty 
 of accurately locating the center of such a faint object, the 
 caution with which apparently confirmatory evidence should 
 be regarded, can be better understood. 
 
 In any given latitude the effect of differential atmospheric 
 absorption of light will always have a tendency to apparently 
 shift the center of illumination towards the zenith of the place. 
 In northern latitudes (greater than 23.5) the apparent dis- 
 placement due to atmospheric absorption will always be greater 
 in June than in December, and about the same in March and 
 September. 
 
 Since there appears to be no marked parallax for the Gegen- 
 schein, and since the absence of strong solar light within the 
 Earth's shadow appears to have no sensible effect on the central 
 portions of this illumination, it follows that much the greater 
 portion of the matter composing the streams must be much 
 farther away from the Earth than the length of the Earth's 
 shadow. 
 
 Further observations, and a much more extended comparison 
 between theory and existing observations, are of course neces- 
 sary before definite conclusions can be drawn. In a general 
 way, however, I have, as it appears to me, conclusively shown 
 that a phenomenon similar in form and position to the 
 observed Zodiacal Light and Gegenschein must necessarily be 
 produced by ejected particles whose orbits have a major axis 
 greater than the Earth's radiusrvector and whose theoretical 
 perihelion is near the Sun's center. 
 
 FORWARD DRIFT OF THE SOLAR SURFACE. 
 
 The planetary perturbations in longitude will always be such, 
 that while the eccentricity of the orbits of some of the particles 
 will be increased (even to the extent of causing a retrograde 
 motion), by far the greater number, especially those having 
 small latitudes, will become less eccentric, so that the resultant 
 effect is to cause the returning masses to strike the Sun with a 
 
By J. M. Schaeberle. 51 
 
 greater angular velocity than that of the Sun's surface, thus 
 causing an actual forward drift of this surface, which will be 
 greatest in the Sun's equatorial regions, in agreement with ob- 
 servation. (A variation in the angular distance of the Gegen- 
 schein from the Earth's radius- vector produced is also a function 
 of the planetary perturbations in longitude.) 
 
 TERRESTRIAL MAGNETISM. 
 
 We know from actual observation that the atmosphere of our 
 planet is continually bombarded by meteoric matter, which is 
 often volatilized before it reaches the Earth's surface. If we 
 imagine the Earth to be in the center of a stream of uniformly 
 distributed matter having a rectilinear motion in a given direc- 
 tion, it is at once evident that owing to the rotation of the Earth 
 the maximum amount of meteoric matter will fall at the place 
 which has the least linear velocity for a given inclination of 
 the local horizon to the direction of the stream's motion, so that 
 at the equator the amount of matter per unit of the area will 
 be at a minimum, while a very decided maximum will be at 
 each pole. Even for the case when the path of the stream is 
 parallel to the Earth's equator, the area about the poles will, on 
 account of the Earth's mass, still receive more foreign matter 
 than a corresponding area nearer the equator, since particles 
 will not only be deflected from the more distant portions of the 
 stream so as to strike the areas near the poles, but the new instan- 
 taneous orbits described by these more distant particles will be 
 such that even the region beyond the poles will be bombarded. 
 If either pole is inclined towards the direction from which the 
 stream is coming, the amount of matter received at that pole 
 will be enormously greater than it will for a corresponding area 
 at the equator. If the surface on which the matter falls is per- 
 manent, there will be a continual accumulation of such matter; 
 if, on the other hand, the surface is a movable one, like water 
 or a moving field of ice, the matter does not necessarily accu- 
 mulate at the place, since ocean currents, sooner or later, will 
 cause a general distribution over the globe. 
 
 Now, if we look at a map of the Earth, we. find that, beyond a 
 certain distance, the north pole is almost completely surrounded 
 by land; the Arctic Ocean bounds the northern coast of Europe, 
 5 s 
 
52 A Mechanical Theory of the Solar Corona, 
 
 Asia, and Alaska. But from Eastern British America towards 
 the pole the whole area is practically land-locked, so that this 
 area will contain the greatest amount of accumulated meteoric 
 matter. The most northerly portions of Siberia will come next 
 in order. At the south pole practically the whole Antarctic 
 region is still unexplored; but while the areas of maximum 
 amount of accumulated matter cannot be accurately located, it 
 is at once evident that the total area over which the mass of 
 meteoric matter can accumulate will be much greater in the 
 Northern than in the Southern Hemisphere. 
 
 Now, the larger meteoric masses which reach the Earth's sur- 
 face are found, on examination, to be strongly magnetic, and 
 the smaller masses are undoubtedly magnetic also. Hence, the 
 arrangement of individual meteoric particles covering a given 
 surface will be such that each one naturally assumes a fixed 
 position with reference to the resultant lines of magnetic force. 
 Of the two poles formed, the one in the Antarctic region will 
 (in agreement with observation) show the greatest force, since 
 the space into which the lines of force are condensed will be 
 least in the Southern Hemisphere. As is also well known from 
 actual observation, there are two areas or centers of maximum 
 force near each magnetic pole; at least such is the case for the 
 Northern Hemisphere, where the area of greater maximum of 
 force is in Northern British America, while the lesser maximum 
 is in Northern Siberia. A possible shifting of the immense ice 
 fields at both poles, of course involves a corresponding move- 
 ment of the meteoric matter imbedded in these fields, and a 
 consequent shifting of the centers of force results. The direc- 
 tion of the Earth's magnetic lines of force should, according 
 to this theory, be largely influenced by the size, outlines, and 
 locations of the continents of the globe; and a summary inspec- 
 tion of a magnetic chart of the globe indicates that such is 
 actually the case. 
 
 THE AURORA. 
 
 Let us now consider the phenomena produced by the passage 
 of the Earth through a particular one of the streams ejected 
 from the Sun. In certain longitudes, at a given instant of time, 
 the direction of the stream will be more nearly at right angles 
 to the direction of the Earth's magnetic lines of force than in 
 
By J. M. Schaeberle. 53 
 
 other longitudes. Now, in the case of a permanent magnet, 
 wound by a coil of some conducting material in the form of an 
 insulated wire, an electric discharge can be produced between 
 the terminal points of this wire, if a magnetic body is moved 
 rapidly across the lines of force of the magnet; the discharge 
 will take place along the line of least resistance. . Applying this 
 principle to the case of the Earth and the stream, then, if my 
 reasoning is sound, it would seem to follow (1) that there will 
 be a tendency for a discharge to take place between the particles 
 of the stream, and that (2) such a discharge will be in the direc- 
 tion of the Earth's magnetic meridian; since each individual 
 particle of the stream, on entering the Earth's magnetic field, will 
 at once take such a position that the line joining its own poles 
 will lie in the direction of the magnetic meridian, and as these 
 same particles will act as conductors, the lines of least resist- 
 ance will be formed in the magnetic meridians. 
 
 If we take it for granted that there is an electrostatic condi- 
 tion in which a positively electrified stratum is separated from 
 one negatively electrified by an insulating stratum of air, then 
 a much simpler explanation results. The lines of least resist- 
 ance will be formed precisely as in the first hypothesis; these lines 
 joining two differently charged strata will evidently be shortest 
 (and the discharges therefore most brilliant) at the magnetic 
 poles, while near the equator no discharges can take place, since 
 any given line of least resistance will lie wholly within a given 
 stratum. The intensity of the discharge will decrease as the 
 area over which the discharges take place increases. When 
 the number of discharges is very great, we have the condition 
 of a great number of nearly parallel luminous lines more or 
 less inclined to each other, so that in addition to the apparent 
 motions towards the magnetic zenith, concentric arches (rays) 
 may be formed, which will slowly vary with variations in the 
 arrangement of the luminous discharges. (See Plate VIII., 
 Figures 10 and 11, inverted. The parallactic effect will depend 
 on the distance of the display from the observer, so that the 
 arches (rays) formed may be either convex or concave in per- 
 spective.) Many of the rapid motions may be only apparent, 
 and due to an actual shifting of the points of perspective 
 intersection of continuous discharges, which have a slight 
 lateral motion. Figure 1, Plate VII., illustrates the low auroral 
 
54 A Mechanical Theory of the Solar Corona, 
 
 arch. If the gratings are slightly shifted, this luminous arch 
 can be changed to a dark segment. Dark segments at other alti- 
 tudes are illustrated in Figures 10, 11, 12 5 and 13. 
 
 Gravitational and atmospheric disturbances will constantly 
 tend to break existing conductors, and to form new ones so long 
 as the suspended particles are sufficiently numerous. 
 
 To further test this theory, the following conditions should 
 be fulfilled: 
 
 As the streams ejected from the Sun have the greatest density 
 in about 15 heliocentric latitude, the number of auroras 
 should be greatest in the months corresponding to the times 
 when the heliocentric latitude ft of the Earth is greatest, and 
 least when latitude of the Earth is zero. As, however, a return- 
 ing stream is more likely to pass near the Earth than an out- 
 going one (on account of the tendency of the outer planets to 
 diminish the latitudes of all the streams), the pole which is 
 turned towards a returning stream when the Earth is in the 
 plane of the Sun's equator will have more decided displays 
 than the one turned towards the Sun. The Earth is in the 
 plane of the Sun's equator in June and December, and during 
 the latter month the north pole attains its greatest inclination 
 away from the Sun, so that the frequency of auroras should be 
 more decided in the Northern Hemisphere in December than in 
 June; the effect of the longer nights will also cause an apparently 
 greater number of displays in December. The Earth attains its 
 greatest heliocentric latitudes in March and September, and both 
 poles are equally exposed to the solar influences at these times, 
 consequently the auroras should not only be most numerous at 
 these times, but in both months the number should be about 
 the same. I take the following data from the article on Meteor- 
 ology, Vol. XVI. of the Encyclopaedia Britannica: 
 
 MONTHLY FREQUENCY OF EUROPEAN AURORAS. 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 April. 
 
 May. 
 
 June. 
 
 July. 
 
 Aug. 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 229 
 
 307 
 
 440 
 
 312 
 
 184 
 
 65 
 
 87 
 
 217 
 
 405 
 
 497 
 
 285 
 
 225 
 
 If a series of observations should be manufactured to fit this 
 theory, the above figures could hardly be improved upon. 
 
By J. M. Schaeberle. 55 
 
 I have not been able to find a series of observations for the 
 Southern Hemisphere. It at once follows that if the streams 
 have periods of maxima and minima of magnitude and fre- 
 quency, the auroras will have similar and nearly coincident 
 periods, the returning streams, as a rule, governing the magni- 
 tude of the display; although, when the perturbations of the 
 streams near the Sun's surface are greatest, the slightly deflected 
 outgoing streams, from the higher latitudes, will be more apt to 
 cross the Earth's path. If these streams are composed of small 
 particles, the auroral phenomena will be largely confined to the 
 higher regions of the Earth's atmosphere. (So far as the final 
 precipitation of such particles to the Earth is concerned, the 
 prevailing direction of the upper atmospheric currents will have 
 much to do with the final location of such masses on the Earth's 
 surface, and here, again, the tendency will be to carry these 
 masses towards the poles,*if our present views on the currents 
 of the upper atmosphere are correct.) 
 
 The monthly variation in auroral frequency is thus accounted 
 for. The variation in the absolute numbers of the auroras, tak- 
 ing the whole world together, will depend upon the number of 
 the streams, and will, therefore, pass through a cycle every ten 
 or eleven years, precisely as the Sun spots do, according to the 
 theory, and in fact. 
 
 VARIABLE AND NEBULOUS STARS. 
 
 It seems to me that many of the irregular variations in the 
 physical appearances of certain classes of stars can be satis- 
 factorily explained by the theory of ejected streams colliding 
 at certain intervals with the returning ones. Viewed from 
 great distances such stars would appear to be surrounded by 
 nebulous envelopes of varying brightness, and according to the 
 position of the axis of rotation and the zones of maximum 
 activity with reference to the line of sight the nebulosity 
 would appear to be either circular or more or less elliptical in 
 outline. The bearing of this theory on Professor LOCKYER'S 
 Meteoritic Hypothesis is significant (so far as I can learn from 
 mere notices of his work" just issued), since these results have 
 been obtained without any preconceived notions as to whether 
 the explanation of certain phenomena would agree or disagree 
 with any other theory. 
 
56 A Mechanical Theory of the Solar Corona. 
 
 COMETS. 
 
 The hypothesis, favored by some astronomers, that the matter 
 now revolving about the Sun, in cometary orbits, was once 
 ejected from the Sun, is, according to the Mechanical Theory, 
 rendered extremely probable, and it would not be difficult to 
 bring forward many strong arguments to support this view and 
 to account for many apparent changes of form in cometary 
 matter. But the length of this postscript has already far 
 exceeded my original intentions, and I now await the result of 
 the final verdict as to the merits of my investigations, to be 
 given by those who are most competent to pass judgment. 
 
Photomount 
 Pamphlet 
 
 Binder 
 Gaylord Bros. 
 
 Makers 
 Stockton, Calif. 
 
 PAT. JAN. 21. 1908 
 
 "36351 
 
 0.652? 
 
 UNIVERSITY OF CALIFORNIA LIBRARY