Show CELESTIAL MOTION PROF CHAMBERLIXS LECTURE BEFORE BE-FORE MATHEMATICAL SOCIETY R J Smltli on the Origin of Symbols Sym-bols of Addition and Subtraction Positive and Negative Quantities I Quanti-ties ten The Utah Mathematical society met in the University on Friday evening President Kerr presiding Professor W 1 Chamberlin was announced as the lecturer of the evening even-ing and on the subject of Celestial Motion he spoke as follows j The solar system as observed by us consists of the sun as a central body around which revolve the planets with their moons a satellites the asteroids aster-oids the periodic omets and an Unknown un-known number of meteor swarms These are permanent members Frequently other comets enter the system pass around the sun then into in-to space never to return The orbits or paths in which the planets asteroids periodic comets and meteor swarms revolve are ellipses which are curves like those made by the intersection of a plane and the surface sur-face of a cone the plane not cutting the base of the cone The natural shape of the orbit of a comet on first entering enter-ing the solar system is a parabola which is the curve made by the intersection inter-section of a plane and the surface of a cone when the plane is parallel with one side of the cone Under certain circumstances the orbit of a comet is an hyperbola which is the figure made by the intersection of a plane and the surface of a cone when the cutting plane passes through the base of the cone and is not parallel with any cage of the cone In order to explain the positions of these orbits relatively to each other we must have some determinate thing to which we may refer all orbits The object of reference used is the plane of the earths orbit called the ecliptic The orbits of the different members of the solar system are inclined to this plane at various angles With few exceptions ex-ceptions the inclination of the orbits ceptons of the planets satellites and asteroids is small many almost coinciding with smal the ecliptic The orbits of the comets and meteor swarms are inclined to this plane at all angles The planets all revolve from west to east the satellites revolve in the same direction as a rule the comets and meteor me-teor swarms pass around the sun in any direction whatever The longest line which can be drawn through the centre of an ellipse is called the major axis on this line are two points called faci so placed that the caled faco sum of their distances from any point in the curve is always the same The these ratio of the distance between two points to the major axis is called In the the eccentricity of the ellipse case of the most of the planets the distance dis-tance between the face is small and the orbits represented on a small scale can hardly be told from circles The orbits of the periodic comets and some or the asteroids have a great eccentricity and hence are very much elongated ver We are now prepared to consider the causes for the motion of bodies in these various ways Up to the time surrounded of Newton great mystery motion It was observed that planetary moton I wa all bodies moving at the earths surface sur-face came to rest in time and from the fact that the motion of the planets was perpetual it was thought that they must be operated upon by causes unknown un-known to us It was not until Newton gave to the world his three laws of motion that mo men began to compare planetary tion with motion at the earths surface wih moton From the first law bodies continue in a state of rest or of uniform motion in a straight line unless compelled to change that state by an external force it was concluded that when a body described a curve it must be acted upon by two5 forces one of projection twc the of and one of deflection In case the planets the deflecting force was seen to be in some way connected with revolve the body about which they From the second law of motion change of motion is in the direction of the force impressed and is proportional to it and the third law action and reaction re-action are eQual and opposite in direction direc-tion it was possible to calculate the intensity of the above forces in any case In general the establishment of the above laws made it possible to calculate cal-culate the motion of any system of bodies when the forces which acted upon them were known and to define requisite to produce what forces were requisie any given motion The problem what force will make a planet describe an ellipse around the sun having the latter in one focus is now to be considered By modern meth ode the solution is a simple one but to Newton the one who first solved i it was quite a difficult one He attacked at-tacked it by starting from gravitation a force familiar to all which causes any body unsupported to fall towards the center of the earth This force extends ex-tends to the tops of the highest mountains moun-tains and beyond them Why may it not be the same force which causes the moon to deviate from a straight line and continually approach the earth Newton believed that the continual con-tinual falling of the moon toward the earth was due to graviation and proceeded pro-ceeded to calculate the intensity of gravity at the distance of the moon which he did by comparing the distance dis-tance the moon falls in one second with the distance objects at the earths surface fall in the same time From this calculation he concluded that the force of gravity varied inversely as the square of the distance from the earths center I From the fact that objects of different differ-ent mass are equally acceleratd at the earths surface by gravity he saw that this force varies as the mass of the attracting at-tracting bodies Assuming the law of force suggested in this way to be universal it was applied ap-plied to the calculation of the Orbits of the bodies revolving about the sun and all calculations based on the law were found to agree with observation and so i was adopted It was easily shown I by the calculus that when a body is projected into space and is attracted by a body according ac-cording to the law its orbit about the attracting body is one of the conic sections before described the particu lar curve of the path depending upon the distance and mass of the attract ing body and the velocity of projection only and if these can be determined the path is known Being acquainted with simple properties of the conic sections we can determine what values must be given to the above quantities in order that the path of a bdy quantties may be an ellipse 3 parabola or a hy perbola The shorter the distance the greater the velocity must be in order that the orbit shall be an ellipse this explains why the nearer the orbit of a planet is to the sun the greater is its velocity and the nearer a planet plaet in the same orbit is to the sun the greater its velocity I the velocity of any planet were to increase by about four tenths of its present preent amount it would move in a parabola and leave our system The velocity acquired by a body falling veloity the rest at an infinite distance to the i1fnle distnce sun is the velocity necessary for a para bolic orbit The fact that most comets travel fs tvel in parabolas suggests to us the great distances from which they come and to which they return If a comet traveling in such an orbit passes in front of a planet it will 4 have Its velocity retarded by the attraction 1 at-traction of the planet and will move in an ellipse becoming a permanent member of our ystm I a comet in an elliptic orbit or one in a parabolic para-bolic orbit passes behind a planet the latter will accelerate its motion and i will probably move in an hyperbola When the various formulae derived from the law of gravitation are applied to calculate the motion of the planets for a long series of years it is found that the calculated positidn a the planet varies slightly from its observed position I may be observed In advance ad-vance behind or aside from its place calculated on the theory of elliptic motion This is due to the fact that the heavenly bodies disturb each other in their motion about the sun and so derange the elliptic forms of the orbits or-bits These variations due to the acton ac-ton of the planets are called perturbations perturba-tions they are extremely small and their magnitudes exactly accord with the calculations effected from the adoption adop-tion of the law of gravitation Corrections are made for these perturbations per-turbations in this way Were the disturbing dis-turbing forces to cease acting the path is known to be a conic section r the effect of such member of the solar system sys-tem supposing all the others not to exist is then added to the suns effect and thus gives the exact motion These calculations are made with such accuracy that after the planet Uranus was discovered a set of perturbations per-turbations observed in the planet which could not be accounted for by the action of any known planet led to the discovery of the outer planet of our system Neptune So accurate were the calculations of two mathematicians mathematic-ians that the exact position in which theplanet would be at a certain time was given before it had ever been seen by man After the lecture Mr R V Smith gave a short history of the origin of the symbols of addition and subtraction subtrac-tion tracing their development from mere signs of relation to the signifi nation whiTh fhcv nnw have He was followed by an excellent vocal duet by Misses Price and Bates Miss Dora Bauman then explained very clearly the doctrine of positive and negative quantities after which the society adjourned |