GIFT OF 
 Mrs. F.J.B. Cordeiro 
 
 r 
 
PUBLICATIONS BY THE AUTHOR 
 
 SPON & CHAMBERLAIN, NEW 
 
 YORK 
 
 [EIGHTS. 
 
 BAROMEIRICAL DETERMINATION OF B 
 
 
 30.50 
 
 THE ATMOSPHERE. 
 
 
 A Manual of Meteorology 
 
 32.50 
 
 THE GYROSCOPE. 
 
 
 Theory and Applications 
 
 31.50 
 
 MECHANICS OF ELECTRICITY. 
 
 
 What is Electricity 
 
 31.25 
 
 PRINCIPLES OF NATURAL PHILOSOPHY. 
 
 32.50 
 
PRINCIPLES OF 
 NATURAL PHILOSOPHY 
 
 BY 
 
 F. J. B. CORDEIRO 
 
 AUTHOR OF 
 'MECHANICS OF ELECTRICITY," "THE GYROSCOPE," 
 'THE ATMOSPHERE," "BAROMETRICAL HEIGHTS," ETC. 
 
 NEW YORK: 
 SPON & CHAMBERLAIN, 123 Liberty Street 
 
 LONDON: 
 E. & F. N. SPON, Limited, 57 Haymarket, S.W. 
 
 1917 
 

 --<? 
 
 \r 
 
 Pinkham Press 
 Boston 
 
PREFACE 
 
 Natural philosophy is the study of nature, or the uni- 
 verse. Experience shows that all effects are preceded by 
 causes, and under identical circumstances all changes are 
 always effected in the same manner. Nature is in a con- 
 tinual state of flux, this flux consisting of a ceaseless inter- 
 change of energy between its various parts, but its action 
 is always uniform and unique. It is this in variableness and 
 uniformity which renders possible a philosophy of nature. 
 The investigation of the manner or laws under which 
 changes take place is comprised under such names as 
 Physics, Dynamics, Mechanics, or Natural Philosophy, 
 but since in this book we shall not confine ourselves merely 
 to the investigation of such processes, but shall consider 
 ourselves free to speculate upon them, insofar as that is 
 possible, the latter term is preferable. It was used by 
 Newton in this sense. 
 
 The ultimate, or primordial causes of nature must ever 
 remain unknown to us, since the infinite cannot be grasped 
 by finite minds, but a certain proximate region has been 
 explored and it is to the credit of mankind that its borders 
 are being continually extended. No important result 
 can ever be achieved without serious and concentrated 
 thought and close and careful reasoning. Such a method, in 
 any of its forms, constitutes mathematics, and any man 
 who reasons carefully and accurately is a mathematician. 
 The theory of changes was called by Newton, its dis- 
 coverer. Fluxions, symbolizing thereby the fluxions of 
 nature. It is now known as The Calculus, and it is only 
 decent that every man with more than a rudimentary 
 education should know the calculus and thereby some- 
 thing of the universe in which he lives. It is the ability 
 and willingness to think which has raised civilized man 
 
 TZ>^' i '-f   i «-, 
 
vi PREFACE 
 
 from the savage, and it is our knowledge of nature and 
 natural forces which determines our present, or any- 
 future civilization. 
 
 The study of such a subject must react upon the student 
 in forming within him a new realization of what truth 
 actually is. He will find that many of his most cherished 
 beliefs would not stand before the Court of Nature, or 
 for that matter before an ordinary law court. For ages man 
 has seemed incapable of distinguishing between the true 
 and false, and has had little desire to do so even when the 
 means were at hand. The idea prevailed, and still prevails, 
 that truths could be created by authority. A statement if 
 loudly proclaimed and accepted by a sufficient following 
 was, and is, supposed to be true irrespective of any in- 
 herent quality. A man may say that he believes and ac- 
 cepts as a truth something which he does not understand, 
 but unless he clearly recognizes for himself the reasons 
 why a thing must be true, it is not a truth for him. The 
 student of mathematics learns from the beginning that 
 nothing but the truth can ultimately prevail and that what 
 is false necessarily carries with itself its own annihilation. 
 Authority has no place in Science, for its results do not 
 rest, or need to rest, upon any personal sponsor, no matter 
 how distinguished, but solely upon the truth or falsity of 
 the reasoning by which they are derived. 
 
 Text books on this subject have been overfull of prob- 
 lems concerning rods and strings and flies walking on 
 circular wires or perfectly smooth tables, so that it is 
 perhaps natural that the impression has prevailed that 
 instead of being an instrument for the acquisition of 
 truth, mathematics are chiefly an agglomeration of symbols 
 by which fantastic results, having no human or practical 
 interest, are obtained. But natural philosophy is not the 
 study of rods and strings. It is the story of the universe. 
 For this reason we shall confine ourselves to processes 
 which are actually occurring about us all the time — to our 
 earth, and to our solar system. 
 
PRINCIPLES OF NATURAL PHILOSOPHY 
 
NATURAL PHILOSOPHY 
 
 1. The Fundamental Law 
 
 The first efforts of man to understand the universe were 
 by metaphysical processes. Without examining the uni- 
 verse as it is, they sought to evolve its plan out of their 
 inner consciences. They conceived in their minds a uni- 
 verse as they thought it might be, or could be, or should be, 
 and sometimes by deduction sought to fit their theory to 
 the case. Generally they did not trouble themselves to go 
 so far as this, but rested content with their imaginary 
 universe. The method consisted of assuming causes 
 instead of interrogating nature herself. They further 
 exhibited the strange tendency of mankind to imagine that 
 by the use of sufficient meaningless words, definite 
 thoughts could be expressed. There were necessarily as 
 many systems as philosophers. But truth is unique, so 
 that only one of all these systems could possibly be true, 
 while the a priori probability was that all were false. 
 In the middle ages when there was no science (knowledge) 
 it was natural that men should have exercised their minds 
 with "Beating the air," but that such methods should 
 persist to the present day is an anachronism. Sooner or 
 later these relics of mediaevalism must disappear. Among 
 certain workers on the present borderland there is a tend- 
 ency to revert to these unsound methods, and within 
 recent years a new kind of metaphysical physics has been 
 developed. The student of natural philosophy must 
 carefully avoid the unclean thing. 
 
 We recognize that the universe consists of matter 
 although we do not know what matter is, or when or how 
 it came into existence, or that it did come into existence. 
 
 1 
 
2 NATURAL PKILOSOPHY 
 
 We simply know that it is there. We further recognize two 
 general forms of matter — ordinary gross matter and 
 the ether. Gross matter is of various kinds — some 70 odd 
 elements — while the ether is apparently uniform, ex- 
 tending through all space, and in fact occupying all space 
 with the exception of that occupied by the atoms of 
 gross matter. We further recognize that none of this 
 matter is ever at rest, but that it changes its position rela- 
 tively to space incessantly — or it is always in motion. 
 We recognize therefore in the universe, matter and motion, 
 and these are the only elements of which we have any 
 cognizance. 
 
 The ether is the connecting medium of the universe, 
 through and by means of which, gross matter imparts its 
 motion to other particles at a distance. We shall see that 
 matter in motion results in a conception called force, 
 and also in a conception called energy, and that these three 
 inseparable entities — motion, force, and energy — are 
 transferred to distant points through the medium of the 
 ether. If two particles of matter were separated by an 
 absolute vacuum, i.e., a space containing no matter of 
 any kind, it would be impossible for the motion of one 
 particle to impress itself upon another, or to influence it 
 in any way. It is further evident that, since matter is 
 neither created nor destroyed, and since motion in a body 
 only arises after it has been transferred to it from some 
 other body — the body imparting the motion losing as 
 much as it transfers — the total amount of matter and 
 motion in the universe must always remain constant. 
 
 We know very little of the ether except that it is a fluid 
 of extraordinary tenuity and under a very high pressure. 
 What its density and pressure are, we know only roughly, 
 but we know that it is a kind of matter, since it possesses 
 the fundamental property of all matter, viz., inertia. 
 Inertia means literally the helplessness of matter, or the 
 inabiUty of matter, from any virtue within itself, to ac- 
 quire motion, or when in motion to bring itself to rest. 
 
THE FUNDAMENTAL LAW 3 
 
 Some external 'motion or force is necessary to produce 
 motion or a change of motion in matter. The pressure of 
 the ether, which is a force, must be due to some kind of 
 internal motion in the ether, but of the nature of this 
 motion there is, as yet, hardly a surmise. The atoms of 
 gross matter take up and reflect this motion of the ether 
 and thus become points or centres from which disturbances 
 radiate in all directions. Every atom therefore influences 
 every other atom in the universe, and the observed 
 attractional and repulsional effects are due to these radia- 
 tions.* The reservoir of motion (energy) is therefore the 
 ether, of which it contains an infinite amount. It has been 
 surmised that the ether is the ground stuff out of which all 
 gross matter has been formed by compression — the 
 density of an element varying with the conditions of 
 pressure, and possibly temperature, under which it was 
 formed. But of such matters we know little or nothing. 
 It has also been suggested that inertia is a property of the 
 ether alone and that gross matter in an ether vacuum 
 would yield to a push without any resistance and cease its 
 motion instantly when the push was removed. 
 
 Any motion of a body necessarily sets up a disturbance 
 in the ether and it has been suggested that the reaction 
 against the body of such a disturbance might account for 
 the resistance to motion which we call inertia. But granting 
 such a possibility, we have no explanation of the inertia of 
 the ether. Inertia therefore is at present the one great 
 mystery of nature. It is not enough to say that it is an 
 inherent property of all matter, for there must be some 
 cause. To move a large body — isolated in space — re- 
 quires a very considerable force and it yields to this force 
 only slowly and reluctantly, but when once in motion 
 there is an equal difficulty in bringing it to rest. 
 
 Experiment shows that the resistance to motion of all 
 
 *The author has shown with some probability, that attraction 
 and repulsion are due to longitudinal waves in the ether, v. "Mechan- 
 ics of Electricity." 
 
4 NATURAL PHILOSOPHY 
 
 matter is proportional jointly to the mass, or quantity of 
 matter, and to the acceleration, or rate at which the 
 motion changes. Naturally two identical bodies would 
 offer twice the resistance of a single body, and when no 
 forces are applied we should expect the state of motion, or 
 velocity, to remain unaltered both in amount and direc- 
 tion. It is only when the state of motion is being altered 
 that we should expect to meet with any resistance, and 
 such we find to be a fact. 
 
 The simplest relation between change of motion and 
 resistance to motion is one of simple proportionality and 
 experiment proves that this is the relation, or the resistance 
 is proportional to the rate at which the motion changes. 
 If we denote the resistance by r, this is expressed by the 
 
 dv 
 relation, r = —m-r- We define a force as the product of a 
 
 mass by its acceleration. To change the motion of a body, 
 
 therefore, we shall have to apply a force, /, equal and 
 
 dv 
 opposite to r, and we have the equation /=wt- (1), 
 
 dv 
 where/ and -r- are directed quantities, or vectors, having 
 
 the same direction. Equa. (1) is Newton's second law of 
 
 motion which he stated "Change of motion is proportional 
 
 to the applied force and such change is effected in the 
 
 direction of the force." It also contains his third law, which 
 
 is, "The action of the force is equal and opposite to the 
 
 reaction (of the inertia)." His first law is merely a corollary 
 
 of the other two. 
 
 Equa. (1) may be called the Fundamental Law of the 
 
 universe, for from it we shall derive all the principles and 
 
 laws of natural philosophy. There is nothing in it which is 
 
 axiomatic or a priori evident, although Newton's laws have 
 
 been called "Axiomata Sive Leges Motus." 
 
 dv 
 But the fundamental law,/ = w -r- , is not actually a law, 
 
 or it does not express exactly the relation between the 
 
THE FUNDAMENTAL LAW 5 
 
 variables. It is merely the statement of experimental 
 results under certain limited conditions, and even under 
 these conditions the law gives the relations only to a very 
 close approximation. If a force acting upon a body at 
 rest in the air imparts a certain acceleration in unit time, 
 the force will have overcome what we call the inertia of 
 the body and in addition a very slight resistance from the 
 air, provided the force is small. Applying the force continu- 
 ously we find that the increment of the acceleration is not 
 constant, but is a function of the previously existing 
 velocity. A curve expressing the relation between the 
 force and the acceleration will be nearly a straight line 
 at the beginning, as it would be for any continuous func- 
 tion expressing a relation. But in our present case the curve 
 will fall away because the resistance of the air increases 
 with the velocity. At a certain velocity the resistance is as 
 the square of the velocity, and when the body has a 
 velocity beyond that with which a disturbance travels 
 in the air — about 1100 ft. per second — a vacuum forms 
 behind the body, because the body moves at the same rate 
 as the vacuum tends to close up. A constant force equal to 
 the pressure of the air would therefore give no acceleration, 
 but merely maintain the velocity constant. 
 
 Let us now repeat our experiments in an air vacuum. 
 We now find that the curve is very nearly a straight line, 
 or the resistance of the inertia is very nearly proportional 
 to the acceleration. But the body is moving in the ether 
 and any motion in a medium is resisted. The velocity of a 
 disturbance in the ether is about 186,000 miles per second, 
 so that if a body were moving with a greater velocity than 
 this it would require a force equal to the pressure of the 
 ether to maintain this velocity constant without imparting 
 any acceleration. Now the greatest velocity in gross matter 
 which has ever been observed is about 300 miles per 
 second. This is insignificant in comparison with the dis- 
 turbance velocity of the ether, and we are therefore justi- 
 fied in concluding that our law — verified by experiment — 
 
6 NATURAL PHILOSOPHY 
 
 holds to an extreme degree of approximation for all matter 
 moving with ordinary velocities. But we cannot apply the 
 formula to matter moving with extraordinary velocities. 
 There are certain bodies called electrons, which are 
 possibly ethereal vortical rings of the smallest size pos- 
 sible (without a lumen), which may move with a velocity 
 approaching an ether wave. As such bodies meet with a 
 resistance which increases with the velocity, it has been 
 assumed that their mass increased with the velocity, and 
 it has been lightly stated that the mass, or quantity of 
 matter, in all bodies — gross or ethereal — is not constant, 
 but a function of the velocity! This is an example of latter 
 day metaphysical physics. 
 
 The amount of matter in a body is at all times, of 
 course, definite, and can only change by the addition or 
 subtraction of matter. 
 
 The energy of a body in motion, called its Kinetic 
 Energy, is defined as half its mass into the square of its 
 
 velocity, or 7 = -pr- . The work done by a constant force is 
 
 defined as the product of the force into the distance through 
 which it acts in the direction of the force, or W =fs. 
 If the force is not constant, 
 
 J 2 n dv , n w / 2 2 \ 
 
 i^^ = J ^d'r^ ^ J ^^^^ ^ 2 V2 ~ ^ / 
 or the increase of the kinetic energy is equal to the work 
 done upon the body between any two positions. Kinetic 
 energy and work are thus equivalent, and we may define 
 energy in general as that which is capable of doing work, 
 and the doing of work as the transference of energy from 
 one body to another. All such processes are evidently 
 reversible. 
 
 In measuring these quantities certain standard units are 
 selected, usually the gramme for mass, the centimeter for 
 length and the second for time, these units constituting the 
 C. G. S. system. Accordingly the unit of force, called a 
 dyne, is one which acting upon a gramme for a second 
 
 W 
 
CENTRIFUGAL FORCE 
 
 imparts to it a velocity of a centimeter a second. The 
 
 earth's attraction upon a gramme at its surface gives it a 
 
 velocity of about 981 cms. per second, for every second 
 
 that it acts, and the force is therefore 981 dynes. The 
 
 attractional force on a kilogram is 1000 times as much, 
 
 but the acceleration is the same. Hence all bodies fall (in a 
 
 vacuum) in the same time. The symbol g is used to express 
 
 this constant acceleration at the earth's surface. 
 
 -.04 cms. , dv f.o4 J 
 
 g = 981 . f = m-j-=mg=m 981 dynes, 
 
 sec. at 
 
 where m is expressed in grammes. If the earth and moon 
 
 were brought to rest and allowed to fall to the sun from 
 
 the same position, the sun being fixed, they would perform 
 
 the journey together and reach the sun at the same 
 
 instant. The accelerations and hence 
 
 the velocities would be the same at 
 
 all times. 
 
 2. Centrifugal Force 
 
 Let us suppose a particle of mass, 
 m, attached to a string, OP, and 
 moving in a circle about without 
 gravity. In the short time dt, it 
 would, if free, move to A. Calling 
 / the average pull of the string, both 
 in amount and direction, we have 
 
 Fig. 1. 
 
 AB =/ 
 
 2m 
 
 d<p 
 
 vstn-^z-.dt, where d(p is the angle POB, and 
 
 V is the velocity at P. Developing the sine into its series, 
 
 2mv 
 
 We shall generally use the Newtonian notation of a super-dot 
 for a velocity and a double super-dot for an acceleration. At 
 
 /d<p d<p^ \ / d<p^ \ 
 
 It ~ "48"+ ^^^') ^ W2;^l 1 — 24 + ^tc. I- 
 
 the limit where d(p becomes zero,/ = mv<p 
 
 mv^ 
 
 = mp<f^. 
 
 where p is the radius of curvature of the path at any 
 instant. Hence whenever a body is moving in a curved 
 
8 NATURAL PHILOSOPHY 
 
 path its inertia gives rise to a centrifugal force, or force 
 directed away from the centre of curvature, which is equal 
 to its mass into the square of its velocity, divided by the 
 radius of curvature. This force measures the tension of the 
 string in our example. Whenever a body is moving freely 
 in a curved path, as a planet or a projectile, the component 
 of the force acting upon it which is ± to the path must 
 always be equal to the centrifugal force and opposite to it. 
 This component obviously cannot influence the velocity 
 of the body but merely deflects its course. A body thrown 
 into the air describes a parabola, and the equation of the 
 curve referred to its vertex is 4:ay = x^, where a is the 
 distance of the vertex either from the focus or from the 
 directrix. The time of falling from the vertex to the level of 
 
 the focus is t = \   — x at this level is 2a and the constant 
 
 horizontal velocity is V 2ag. At the vertex the centrifugal 
 
 2 fH a 2. 
 
 force is therefore • — This must be equal to mg and the 
 
 P 
 
 radius of curvature at the vertex is 2a. 
 
 3. Principle of Least Action 
 
 We shall now examine the manner in which motions are 
 executed in nature with a view to ascertaining whether 
 any particular economies are effected. When a system is 
 contained in a closed surface across which forces (motions) 
 do not pass, the total energy of the system remains con- 
 stant and the processes consist only of reversible inter- 
 changes between work and kinetic energy within the sys- 
 tem. Such a system is said to be conservative since its 
 total energy remains constant. But if during these changes 
 the work is not transformed wholly into gross motion, or 
 gross kinetic energy, but some part of it is expended in 
 producing fine (molecular) vibrations, such as heat 
 (friction) , which fine motion is transferred to the ether and 
 thus lost to the system, then the system is non-conserva- 
 tive and its energy does not remain constant. 
 
PRINCIPLE OF LEAST ACTION 9 
 
 In a conservative system the kinetic energy of any 
 configuration obviously depends simply upon the position 
 or co-ordinates of that configuration, so that no matter 
 what changes it has undergone, on returning to the same 
 configuration it always has the same kinetic energy. 
 Hence, taking account of friction, the impossibility of 
 perpetual motion in an isolated system. 
 
 Let us suppose that a system changes from a certain 
 configuration at the time h, by its own free motion, to 
 another configuration at the time ^2, the coordinates of 
 any one particle of the system at any instant being x, y, z. 
 If we vary the motion by guides or constraints so that at 
 any instant a particle instead of occupying its natural 
 position X, y, z, occupies a position, x -\- 8x, y -^ 8y, z -{- 8z, 
 where 8x, 8y, 8z are infinitesimal arbitrary quantities, 
 but the initial and final positions are the same, the differ- 
 ence between the work done in the free paths and in 
 the varied paths is 
 
 8W = 1: {X8x + Y8y + Z8z) = 
 
 d (dy ^\ , d (dz . \ (dx , 8x . dy , 8y . dz , 8z\l 
 
 where X, Y, Z are the components of the force acting upon 
 any particle and t is the independent variable. 
 
 The last term is the variation of T, the kinetic energy, 
 
 - -=-f[(sy+(i;+(i:)> 
 
 Integrating, 8 C (w + T\dt = Sm/'^Sx + ^^8y + 
 
 dt^y\h 
 
10 NATURAL PHILOSOPHY 
 
 Since the variations vanish at the upper and lower limits, 
 '{W + T)dt = 0. 
 
 \r 
 
 Negative work, or stored up energy, is called Potential 
 energy, designated by V, and W = — V. 
 
 Hence 8 f V - V)dt = 0. 
 
 ,r<- 
 
 This result is known as Hamilton's Principle. It is entirely 
 general and applies to non-conservative as well as con- 
 servative systems. Since in a conservative system 51^ = 8T, 
 
 d rV + W)dt = 25 r Tdt = 0. 
 
 For the natural paths, therefore, these integrals have 
 a stationary value, or for all other infinitely near paths 
 they are either greater or less. As a matter of fact all these 
 
 integrals are minima, for taking the expression 5 i Tdt = 0, 
 
 it is evident that by causing a particle to execute an in- 
 finitesimal loop at any point of the actual path, the in- 
 tegral must be greater than for the actual path, and the 
 actual or free path cannot be a maximum. 
 
 The integral I Tdt is called the action of the system, 
 
 and we have proved that in a conservative system the 
 natural action is less than that in any other infinitely 
 near path. This is known as the Principle of Least Action. 
 It means that the average work, or kinetic energy, for the 
 time, or the time mean of the work or kinetic energy, mul- 
 tiplied by the time, cannot possibly be made less. Or if 
 we desire to effect a given change in any way different 
 from the natural way, we shall have to expend more 
 energy, or do more work than nature does, provided the 
 change is effected in the same time. 
 
 This is a remarkable result contained implicitly in the 
 fundamental law. 
 
 Let us take the case of a body moving under the in- 
 
PRINCIPLE OF LEAST ACTION 
 
 11 
 
 fluence of gravity between any two points. Taking the 
 vertex of the path as origin, the time as abscissas and the 
 work as ordinates, the action between and 2 will be rep- 
 
 2 ' 
 072 will be a parabola, though 
 bola of the path. The action, 
 
 resented by the area 023. Since W = ^' the action curve 
 
 not of course the para- 
 6 3 
 
 s 
 
 o 
 
 Witi 
 
 . The time 
 
 mean of the work, or the average 
 work for the time is therefore 
 }4 of the total work, and the 
 product of this mean by the 
 time, or the area 023 = 0453 = 
 the action, cannot possibly be 
 made less. If the motion is be- 
 tween two points on the same 
 level, corresponding to 1 and 2, 
 the action is zero, for the action between 1 and is a minus 
 area, since W is negative in this part, and the action between 
 and 2 is an equal positive area. There are two free paths 
 by which a body may move between any two points — say 
 points corresponding to 1 and 7 — but the action in each 
 case will be the same, viz., 076 — 023 and —6723. Between 
 points corresponding to 1 and 2 there are two free paths 
 and an infinite number of possible guided paths, but in 
 every case the action is zero. In this case alone, which is an 
 absolute minimum, it is possible for a guided path to have 
 no greater action than the free path, but in all other cases 
 the free action is less than any constrained action, and 
 in all cases the free action is the least possible. Nature, 
 therefore, takes no care of time but is very exacting as to 
 how her energy shall be expended. She insists that the 
 action shall be the least possible. When, however, no 
 energy is expended she becomes economical both as to 
 
 time and space 
 
 '.r™-?g 
 
 vds = 0. When no energy 
 
12 NATURAL PHILOSOPHY 
 
 is expended, v is constant, and s is a minimum as well as t, 
 for the path between any two points becomes a straight 
 line. A body under no forces, or a ray of light, moves in a 
 straight line, or they traverse the shortest possible path 
 between any two points in the shortest possible time. 
 
 4. Brachystochrone and Tautochrone 
 
 Whenever energy is expended it is easy to improve upon 
 nature in the matter of time, and we shall construct a path 
 
 such that a body moving 
 under gravity may pass from 
 one point to another in the 
 least possible time. 
 -^1 ^-"^^ ^ i^ ^^ .B Let us suppose (Fig. 3) 
 f\ [ \ J that our body is at the point 
 
 \^^V \<>^)jt^ 1 a-nd we wish to conduct it 
 
 ^ to the point 2 in the shortest 
 
 p ^ possible time. It has a veloc- 
 
 ity 2^0 at 1, and we take as 
 
 our base of ordinates a line AB which is ^r above 1. 
 
 2g 
 
 t= I — = I = — dy, where X =--T- 
 
 ■J 
 
 8dx _ dx. x' 
 
 r 6xd. ( — ^' \ 
 
 J VVI -\-x''- yl2gy/ 
 
 The variations vanish at the points 1 and 2, so that for 
 
 ^ to be a minimum, the last term must vanish, or 
 
 x' 1 
 
 ■= must be a constant — say => where 
 
 ^\+x'^ V 2gy 2yjag 
 
 a is a constant to be determined directly. If r is the angle 
 
 which the path at any point makes with the y ordinate, 
 
 x' 
 sin r = — . Hence, the required path, in terms 
 
 of y and r, is 2a sin^r = y. This is the equation of a cycloid 
 
BRACHYSTOCHRONE AND TAUTOCHRONE 13 
 
 where a is the radius of the generating circle and y is 
 measured from its base Hne. The solution is entirely- 
 general and unique, since a circle can always be found 
 which rolling along a given base line will trace with one of 
 its points a curve which shall pass through two given 
 points, and there is only one such circle. The radius of the 
 generating circle is readily found from the data. 
 
 The cyloid has also the property that the time taken in 
 falling from rest to the lowest point, C, of the curve, is the 
 
 ds V 1 4- x'^ 
 
 same for all starting points. For dt = — = dy, 
 
 V ^2g{y-y,) 
 
 where y\ is the ordinate of the starting point. 
 
 t^ {''jA±^dy, 
 yiJ ^2g{y - yi) 
 
 But l+x'^ = ,r^^^—, and t = ('""J ^ = 
 
 2^ - y yj llgiy - y^) {la - y) 
 
 2^ sin "Vf^^r = -V^- 
 J g J2a - y{[yi ^g 
 
 Or the time is independent of the starting point. 
 The time of a complete oscillation is 2x ^— . Since a 
 
 cycloid is the involute of two equal cycloids placed 
 above it as in Fig. 3, or since it can be described by 
 fixing a string of length 4a at D and wrapping it 
 around the upper cycloids, it is evident that small os- 
 cillations of a particle about the lowest point, C, will co- 
 incide very nearly with those of an ordinary simple 
 pendulum of length 4a. Hence the time of a small oscilla- 
 tion of a simple pendulum is 2^-^—, I being its length. 
 
 We can easily derive this tautochronous property of the 
 cycloid in another way. The intrinsic equation of a cycloid 
 is 5 = 4a sin <p, where s is the length of the curve measured 
 from any point and <p is the angle between the directions of 
 the curve at the first and any second point. Measuring 5 
 from the lowest point, since the acceleration along the 
 
14 
 
 NATURAL PHILOSOPHY 
 4a 
 
 curve IS 5 = — g sm ^, 5 = 
 
 g 
 
 5 (1). Such a motion where 
 
 the force of restitution is proportional to the distance from 
 the position of equilibrium is called harmonic motion, and 
 a particle oscillating about the lowest point of a cycloid 
 therefore executes a harmonic motion. Integrating (1), 
 
 -VJ 
 
 (5i2— s2), where si is the starting point. 
 
 t = 
 
 -1 
 
 '4a • "^ ^ 
 ^" sm — 
 
 g ^1 
 
 ^1 
 
 and the complete period is 
 
 = 27r^_, which is independent of the starting point. 
 
 We shall verify the principle of least action by deter- 
 mining directly what the path of a body moving under 
 gravity must be in order that the action 
 between any two points of the curve 
 shall be a minimum. The body is pro- 
 jected from 1 with a velocity Vq and 
 guided (if necessary) along a curve pass- 
 ing through 2, such that the action be- 
 tween these two points is a minimum. 
 Take as origin of y co-ordinates a base 
 
 Fig. 4. 
 
 line OX, such that yi =-— 
 
 2g 
 
 Then r'^dt=~ ^vds— P ^2gy ^l+x'2.dy, 
 
 and this integral must be a minimum. Considering y the 
 independent variable and varying x, we have 
 
 5 f" V2^ Vl -]-x%dy= C^—iM=x'ddx=0=^ 
 ij I J Vl-t-r^'2 
 
 x'8x V2g:vP 
 
 Vl +x 
 For the action to be a minimum, 
 
 '2|l ij \ -Jl+X'2 J 
 
 ^2gy 
 
 ^ll+x'2 
 
 x' must be a con- 
 
CENTRAL FORCES 15 
 
 stant, c. When T = ;r" > "the curve is horizontal and it does 
 2g _ 
 
 not extend above this level since values become imaginary. 
 Let c2 = lag, where a will be determined directly. Hence 
 
 x' = -^ ^ and ^ = 2 Va ^y — a + C. We have not yet 
 jy — a 
 
 fixed our origin of co-ordinates, but shall take it, so that 
 
 C = 0. Hence the required path is x2 = Aaiy — a). 
 
 This is the equation of a vertical parabola with its directrix 
 
 as the X axis, and a the distance of the vertex from the 
 
 directrix, or from the focus. The velocity of any projectile 
 
 at its highest point is thus the same as if it had fallen 
 
 to that point from the directrix from rest. The required 
 
 path of least action is therefore a parabola passing 
 
 through the two points and bearing a certain relation to 
 
 the initial velocity. Since three conditions fix any conic, 
 
 the parabola is determined. But this parabola is precisely 
 
 the free path. 
 
 6. Central Forces 
 
 Let a particle of mass m move subject to a force which 
 radiates from a fixed point. Let r be the distance Jrom the 
 point to the particle at any instant and <p the angular 
 velocity of this radius. The forces acting along a radius are 
 the centrifugal force and the force / radiating from the 
 point. The effective force along a radius is mr, and mr = 
 nir<p2 + / (!)• If/ attracts the particle it is negative and 
 opposes the centrifugal force. Integrating (1) 
 
 ij 1 2 |i 2 1 ij 
 
 mr2(pd(p 
 
 • 2 , mr2<p2 \ 2 ri 
 
 = mr2 -\ ^— \ — \ tp d {mr2<p). The last integral 
 
 1 1 ^ 1 1 1 J 
 
 must therefore be zero, and mr2(p = mrvc is constant, 
 where Vc is the circumferential velocity, or the velocity ± 
 to the radius. It will be noted that the centrifugal force, 
 being an internal force arising solely from inertia, can do 
 no work upon the particle, the increase or decrease of 
 
16 NATURAL PHILOSOPHY 
 
 kinetic energy being due solely to the external force, /. 
 The product of a directed quantity by its distance from an 
 axis is called the moment of the quantity about that axis. 
 The moment of the angular velocity is evidently the 
 linear circumferential velocity. The moment of a force 
 about an axis is called a Couple. If two equal and opposite 
 forces act at right angles to the extremities of a line which 
 is fixed at its centre, the measure of such a couple is the 
 product of one of the forces into the distance between them, 
 or it is the moment of one of the forces about the 
 extremity of the line joining the two forces. The product of 
 a mass by its velocity is called the momentum of the mass, 
 or it is the quantity of motion in a body. 
 
 We have obtained the important principle that a body, 
 under the action of a central force only, preserves the 
 moment of its momentum about the centre constant. 
 
 The area described by a radius is >^ I r^dcp = Ct, and this 
 
 area is proportional to the time, which is Kepler's second 
 law. If the body is subject also to circumferential forces, 
 it does not preserve its moment of momentum constant. 
 For the moment of the elementary circumferential force, 
 
 or the elementary couple, is m -7- (r^cp) ='yyi -rr (^^c)> and the 
 
 time integral of this is mrvc 
 
 We have thus a second important principle — the time 
 integral of a couple about a fixed axis is measured by the 
 increase (or decrease) of the moment of momentum about 
 that axis. Or, since the moment of momentum of a body is 
 its moment of inertia about an axis into the angular velocity 
 about the axis, this is evidently the time integral of the 
 couple acting about that axis. Further, since the kinetic 
 energy about any axis is half the moment of inertia about 
 the axis into the square of the angular velocity, this is 
 evidently the angle integral of the couple. For a particle, 
 w, the couple at any instant is mrDtr<p, and the angular 
 
ATMOSPHERIC CIRCULATION 17 
 
 2 
 
 integral is m I r(pd{r<p) = m—Ty- 
 
 , or the angle integral of 
 the couple is measured by the increase of the kinetic energy. 
 
 7. Atmospheric Circulations 
 
 The following problem is fundamental in the theory of 
 the circulation of our atmosphere, or of any planetary 
 circulation. The question is, what path would a moving 
 mass of air (wind) describe on a rotating spheroid, if 
 such mass were practically unhindered by other masses of 
 air, or by friction. 
 
 Let us suppose a plastic mass rotating about an axis and 
 subject to its own gravitation. If it were not rotating it 
 would assume a strictly spherical form and the lines of 
 force at the surface would all pass through the centre. 
 The centrifugal force of rotation causes the mass to bulge 
 at its equator and transforms the sphere into a spheroid, 
 the lines of force at the surface being, as before, ± to the 
 surface. If co is the angular velocity of rotation and ^ the 
 angle which a radius makes with the equatorial plane, 
 the centrifugal force at its extremity is r cos d^ co^, and 
 the resolved part of this along the surface, towards the 
 equator, is r sin ^ cos t? 0)2 to a close approximation, if 
 the spheroid differs but little from a sphere. For a particle 
 to be in equilibrium on the surface, the resolved part of 
 the gravitational force along the surface, towards the pole, 
 must be exactly equal to the surface centrifugal component. 
 This gravitational component would evidently restore the 
 spheroid to its original spherical form, immediately the 
 rotation ceased. For a particle to be in equilibrium at any 
 point on the surface it must therefore have exactly the 
 rotational velocity, co. If the particle has an angular 
 velocity, ^p, about the axis, the force urging it along a 
 meridian is / = r sin t? cos ^ {oo'^ — yp'^ ) , o, positive value 
 denoting acceleration towards the pole and a negative 
 value towards the equator. The moment of momentum 
 about the axis must remain constant since there is no 
 
18 NATURAL PHILOSOPHY 
 
 couple about this axis, or cos2 t?i/' = C. Hence f = r^ = 
 
 r sin T? cos r? ( co2 — V Integrating and putting r> = t?o 
 
 when t? becomes zero, we have 
 i>2 cos 2t?o - cos 2t? 
 
 in T? cos r? [ co2 — \ Integrating and putting r> = t?( 
 
 Lve 
 ^ . aj2 + O (tan2 t?o - tan2 i^). 
 
 cos 2t?o - cos 2i? = 2 C /"-^ - ^ j 
 
 and tan2t?o - tan^i} = ^ ( lAo - ^Aj- 
 
 Hence t?2 = C (t/'o - ^) A - ^ j (1). There are two 
 
 values of ^ which makes t^ vanish. One of these we have 
 
 already taken as xpo, and the other is -i— The path is 
 
 tangent to a parallel of latitude at these points and the 
 whole curve must lie between these two parallels. Writing 
 the angular velocities at the upper and lower parallels as 
 
 ypu and ^l/o, we have ^^ = -^ {■^p^ _ ^) (^ _ ^^) (2). We 
 
 shall call linear velocities along a parallel, horizontal veloci- 
 ties, and linear velocities along a meridian, polar velocities, 
 
 designated by vu and v^,. Since \l/ = ^ = r» we 
 
 ^ r cos t? cos2 ?? 
 
 have from (2), Vp2 = i- {vh^ - Vo^) {v^^ - Vh^) (3). 
 
 It must be borne in mind that these are absolute hori- 
 zontal velocities — not relative to the earth. Designating the 
 relative horizontal velocity by Vrh, Vrh = r cos t? (co — ^) = 
 
 -rC r^ (i)C ... 
 
 r cos t? CO = vu (4). 
 
 cos ?> Vh 
 
 Let Vr be the total velocity relative to the earth. Then, 
 from (3) and (4), 
 
 ■M^„, ^± r^o o • /- ^0 cos t?o "i^u COS T?„ J 
 
 Now n 0)2 62 = vq^ Vu^, smce C = — = — -, and 
 
ATMOSPHERIC CIRCULATION 19 
 
 co2 = \Po xj/u. Hence Vr^ = vq^ — 2r2coC + ^u^ (5). Or the 
 velocity relative to the earth is constant. It will be seen 
 that the maximum polar velocity occurs on that parallel 
 where there is no poleward acceleration, or where \p = cc = 
 ^\j/o yj/u. This maximum polar velocity is v^ — Vq. Designat- 
 ing the absolute horizontal velocity at this parallel 
 of equilibrium by Ve, ^ = v^ Vq, or the horizontal velocity 
 at this point is the geometric mean of the extreme hori- 
 zontal velocities. Since cos2 t?o^o = cos2 ^u'^u and ^i* = -7-. 
 
 r cos ^Q-j/Q = r cos t?„co, or the absolute horizontal velocity 
 at the lower limit is equal to the velocity of the earth 
 at the upper limit, and the absolute horizontal velocity 
 at the upper limit is equal to the velocity of the earth 
 at the lower limit. The motion is thus completely deter- 
 mined. 
 
 As on all planets the temperature is greater on the whole 
 at the equator and becomes gradually less towards the 
 poles, the atmosphere rises at the equator and is replaced 
 by other portions flowing along the surface from north and 
 south. Such streams, effected by differences of tempera- 
 ture, will be forced to execute such paths as we have just 
 determined, and as in crowds when a general trend is once 
 established there is little mutual interference, so the 
 circulation of the air must approximate closely to the 
 dynamical factors. 
 
 The equatorial circulation of a planet is shown in Fig. 5. 
 The currents flowing towards the equator are at first 
 close to the surface but are continually deflected — to the 
 right in the northern hemisphere, to the left in the southern 
 hemisphere. They do not reach the equator, but are de- 
 flected due west at a high level. At the parallels of equilib- 
 rium, indicated by the dotted lines, the directions are 
 due north and south, the upper currents going poleward 
 while the surface currents are towards the equator. The 
 limiting parallels, north and south, are functions of the 
 difference of temperature between these parallels and also 
 
20 NATURAL PHILOSOPHY 
 
 of the rotational velocity of the planet. Since the curva- 
 ture of the paths is a minimuni nearest to the equator, 
 the general trend of this circulation is constantly to the 
 west. 
 
 There is likewise a fiat polar circulation the extent and 
 characteristics of which are determined to a certain extent 
 by the elements we have just discussed. Between the equa- 
 torial and polar circulations is the temperate circulation 
 composed of several partly independent and not sharply 
 
 
 Fig. 5. 
 
 differentiated zones. The temperate circulation as a whole 
 moves towards the east with varying northerly and 
 southerly components. As Lord Kelvin has pointed out, 
 there is on the whole a slow shifting, due to friction, of the 
 surface currents towards the poles with a counterbalanc- 
 ing slow shifting at higher levels from the border of the 
 polar circulation to that of the equatorial circulation. 
 The circulation of any planetary atmosphere is thus 
 differentiated into six distinct circulations, the borders of 
 which are very sharply marked. The currents of the equa- 
 torial circulation are very constant both as to their in- 
 tensities and the shapes of their paths, the polar circulation 
 less so, while the temperate circulation is still less stable. 
 It is hardly necessary to state that what would be a con- 
 stant and stable condition in all the circulations, pro- 
 vided our postulated conditions existed, viz., that the 
 earth had a homogeneous surface and were symmetrically 
 heated about the equator, does not actually exist because 
 
MOTION OF RIGID MASSES 21 
 
 the inclination of the sun to the equator is constantly- 
 shifting and because the surface is irregularly divided into 
 land and water, the land being of varying altitudes. In 
 the ideal conditions the equatorial circulations would 
 never reach the equator, while in the actual conditions 
 they frequently cross it. This leads at times to cyclones 
 and various other abnormal disturbances, a fuller 
 discussion of which will be found in "The Atmosphere," 
 by the author. It is interesting to note that Dr. Percival 
 Lowell has observed ** Faint lacings. . . criss-crossed 
 by darker lines" in the equatorial zones of both Jupiter 
 and Saturn. It is quite possible that these are cloud 
 streams in their equatorial circulations, and a glance 
 at Fig. 5 shows that they might have just such an 
 appearance when viewed in a telescope.* 
 
 From the law of constant moments of momentum, 
 4/ = 2C sec2 ^ tan M = 2yp tan M, and RcoS'&^ = 2R^ sin t?^. 
 Putting ypr for the relative angular horizontal velocity, 
 or i/'y = ;/' — CO, we have R cos ■&4^ = 2R {xj/r + co) sin ^4, 
 and R^ = —R sin i} cos ^.2o)-j/r approximately, if xpr is 
 small compared with co. Hence we may write approxi- 
 mately, R cos ^^ = 2Roi sin M. Consequently if p be 
 the radius of curvature of the path and Vr the relative 
 
 V ^ 1) 
 
 velocity at any point,— = 2co sin i^Vr, or p = tj ^ — jr. 
 
 p zco sm V 
 
 The curvature of the path is therefore nearly proportional 
 
 to the sine of the latitude and inversely proportional to the 
 
 relative velocity. This result was first given by Ferrel. 
 
 8. Motion of Rigid Masses 
 
 We have hitherto considered the motion of particles, or 
 of masses of matter supposed concentrated into a mathe- 
 matical point. We shall now investigate the motion of 
 masses having definite dimensions. We can consider a rigid 
 body as made up of an infinite number of particles which 
 are held together by an unyielding non-material frame. 
 *Dr. Percival Lowell. Popular Astronomy. April, 1910. 
 
22 NATURAL PHILOSOPHY 
 
 In any field of force every particle will be subjected to a 
 
 force, F, which we shall call the applied force. It cannot 
 
 obey this force, as a detached particle would do, by reason 
 
 of its fixed connections, but the effective force on each 
 
 particle will be the geometrical resultant of the applied 
 
 force and the sum of the reactions of the neighboring 
 
 particles, and the particle will obey this effective force the 
 
 same as if it were free. That is, the actual infinitesimal 
 
 path, ds, will be in the direction of this force, and it will 
 
 (Ps 
 oppose to this force its inertia, measured by m-7— , which 
 
 likewise measures the effective force. Summing all the 
 forces we have three groups — the applied forces, the 
 reactions of the neighboring particles, and the forces of 
 inertia or the effective forces. Now the sum of the reactions 
 among the particles must be zero, since there is no relative 
 motion between them. Hence the geometric sum of all the 
 applied forces must be equal to the geometric sum of all 
 
 the effective forces. Or SF = Sm-r—, where ds is the 
 
 actual elementary path of each particle, and S signifies 
 geometric sum — not algebraic sum. This is D'Alembert's 
 Principle. 
 
 Taking any ± axes, and resolving each applied force, 
 F, into X, y, Z, parallel to these axes, we have 
 
 Putting l^mx = Mx, Zmy = My, Xmz = Mz, where M is the 
 total mass, these equations determine a point in the body 
 having co-ordinates oc, y, 2, and this point is called the centre 
 of inertia, or centre of mass. It is evidently a fixed point 
 in the body, irrespective of whatsoever forces act upon it. 
 The interpretation of these equations is that if we 
 transfer every applied force to the centre of inertia, parallel 
 to itself, the geometrical sum of these forces will be equal 
 to a single force acting upon the entire mass considered as 
 concentrated at this point and this force will, of course, 
 
MOTION OF RIGID MASSES 23 
 
 be equal and opposite to the inertianal force of such a 
 concentrated mass. In other words, the motion of the 
 centre of inertia will be same as the motion of a ma- 
 terial point or particle of mass M, under a force which is 
 the geometrical resultant of all the applied forces acting 
 at this point parallel to their original directions. 
 
 We have next to consider that the applied forces do 
 not act at the centre of inertia, but on the several particles. 
 
 Since sy = Sm-^, l^xY = Sm^c -^ and S^^X = ^my -^, 
 
 xY is, the moment of the force Y about the axis of Z, 
 and yX is the moment of X about this axis. Hence 
 
 i:,{xY— yX) = ^mfx-r^ — y -7^ j means that the couple 
 
 about the z axis due to all the applied forces is equal and 
 opposite to the couple about this axis due to the inertianal 
 forces. We have then, 
 
 ;(xmx\ 
 h^myj 
 
 and 
 
 B. .(,z-.y) = ..(.g-.S) = 
 
 d ^ / dz dy\ 
 
 _ s^^y J, - ^ ^ j 
 
 d -, / dx dz\ 
 
 d2z d2 / \ J2 
 
 dt2 dt2\ J dt2 - 
 
24 NATURAL PHILOSOPHY 
 
 Equas. A state that the sum of the momenta of all the 
 particles in any direction is equal to the component in 
 that direction of the momentum of the total mass moving 
 with the velocity of the centre of inertia. 
 
 Equas. B state that the derivative with respect to the 
 time of the moment of momentum about any axis is equal 
 to the couple about that axis, a result which we have 
 already obtained. It follows that the motion of a rigid 
 body under the action of any forces can always be re- 
 solved into a translational motion of the centre of inertia 
 and a rotation about an axis through that centre. It is 
 further evident that these two motions are entirely in- 
 dependent of each other, so that if we oppose the transla- 
 tional motion, the rotation will occur as before, and if we 
 prevent the rotation the translational motion will be un- 
 influenced. 
 
 We can arrive at these results more simply as follows: 
 The applied forces can be reduced (geometrically) to a 
 single force acting through some line within or without 
 the body (but if without the body to be considered as 
 rigidly connected with it) , and the geometric sum of all the 
 inertianal forces is a single force acting through this same 
 line, but in the opposite direction. Dropping a ± from the 
 centre of inertia to this line, and applying to the centre of 
 inertia a force equal and parallel to the resultant of the 
 applied forces and two forces equal to half this force, but 
 opposite in direction, to the extremity of the _L and an 
 equal distance on the other side of the centre of inertia 
 respectively, this system will be in equilibrium. But this 
 system combined with the resultant of the applied forces is 
 equivalent to a single force acting at the centre of inertia 
 equal and parallel to the resultant, together with a couple 
 about an axis through the centre of inertia. Likewise, revers- 
 ing all the directions, the resultant of all the inertianal forces 
 is equivalent to an equal and parallel force acting at the 
 centre of inertia, together with a couple about an axis 
 through this centre. The axis of the couple is JL to the plane 
 
MOTION OF RIGID MASSES 25 
 
 through the line of action and centre of inertia, and the 
 intensity of the couple is the moment of the resultant of all 
 the applied forces about the centre of inertia. 
 
 If the resultant of the applied forces passes through the 
 centre of inertia there can be no rotation. A homogeneous 
 sphere in a centrally attracting field can acquire no ro- 
 tation and is said to be centrobaric. That is, the resultant 
 line of attraction of an attracting point always passes 
 through the centre of inertia of the sphere. This is evident 
 from symmetry. Likewise no body, whatsoever its shape, 
 can acquire a rotation in a uniform parallel field, such as 
 the field at the earth's surface. 
 
 It is to be observed that generally the motion of the 
 centre of inertia is not the same as if the whole mass were 
 first concentrated into its centre of inertia and then acted 
 upon by the field. What we have proved is that for any 
 field it is the same as if the applied forces were applied 
 to the total mass at the centre of inertia, parallel to their 
 original directions. However, in certain fields the result 
 will be the same in either case. In uniform parallel fields 
 such will obviously be the case, and also {v. Art. 24) when 
 the forces tend to a fixed centre and vary as the distance 
 from that centre. 
 
 If we define the centre of gravity as a point in a body 
 such that when fixed the body is not rotated by the field 
 in any position, it is evident that when there is such a point 
 it is the centre of inertia, but that generally there is no such 
 point. The earth being a spheroid is not centrobaric for 
 central fields and therefore has no such point. The sun's 
 field and the moon's field both produce rotations of the 
 earth which result in the precession of the equinoxes. 
 Usually the term Centre of Gravity is taken as synony- 
 mous with Centre of Inertia. Since every mathematical 
 conception should have a single name and as there are 
 other fields than gravitational, it would seem advisable 
 to employ the term Centre of Inertia alone. 
 
 Using polar co-ordinates, x = r cos i^, y = r sin t?, where 
 
26 NATURAL PHILOSOPHY 
 
 t? is the angle between a radius in the %, y plane and the 
 X axis, 
 
 sint?z?2 J. Hence the couple about the z axis is llmr^d- = 
 
 ^Zmr^. The integral Swr2 is called the moment of inertia 
 of a body about an axis ± to r. Letting Swr2 = Mr2, 
 r is called the Radius of Gyration, and it is the average 
 radius which would give the same moment of inertia if 
 the whole mass were concentrated at its extremity. 
 A couple is therefore measured by the moment of inertia 
 into the angular acceleration about an axis. 
 
 9. Moments of Inertia 
 
 Taking some point in a body as origin of rectangular 
 co-ordinates, let us draw radii in all directions from the 
 origin of such lengths that the moment of inertia about 
 any radius as an axis shall be equal to the square of the 
 
 reciprocal of the radius, or J = — , where I is the moment of 
 
 r2 
 
 inertia. The locus of the extremities of these radii will be a 
 
 surface. Designating the moments of inertia about the 
 
 j_ 
 
 = i^,/,=2m(:^2 + ^2)= 1 
 
 7-22 ' 7-3. 
 
 is 2 Smr2, a constant, where r is the distance of any element 
 from the origin, and ri, r2, rs refer to the momental 
 surface. We have taken any axes, so that our surface has 
 the property that the sum of the squares of the reciprocals 
 of any three X radii is constant. Such a property belongs to 
 an elHpsoid alone. For let ai, jSi, 71, : ^2, ^2, 72; "3, i^a, 73 
 be the direction angles of any three mutually J_ radii 
 referred to the principal axes of an ellipsoid, a, b, c. Then 
 
 1 COS2q;i COS2j8i COS27i 
 
 r^2 = a2 + 62 + ""^T" etc., 
 
 axes as 7*, ly, I^, I^ = Sw (y^ +22) = :r^yly = 2m {x^ + z^) 
 , Iz = 2w {x^ -\- y2) =, ^, The sum of these moments 
 
IMPULSIVE FORCES 27 
 
 Hence, at any point of any body it is possible to con- 
 struct an ellipsoid with this point as a centre such that 
 the square of the reciprocal of any radius is equal to the 
 moment of inertia of the body about that radius as an axis. 
 The ellipsoid corresponding to any point is called the Mo- 
 mental Ellipsoid for that point. The principal axes of this 
 ellipsoid are called the principal axes of inertia of the body 
 for that point. The principal axes of inertia corresponding 
 to the centre of inertia are called simply the Principal Axes 
 of the body, and the moments about these axes are the 
 Principal Moments of Inertia. Generally the three principal 
 moments of inertia have different values and such bodies 
 are said to be triaxial. When two of the moments are equal, 
 the body is biaxial and when all three are equal the body is 
 uniaxial. 
 
 Taking any axis about which we wish to find the mo- 
 ment of inertia as the z axis and x, y as the co-ordinates of 
 the centre of inertia and x^, y^ as the co-ordinates of an 
 element referred to parallel axes through the centre of 
 inertia, since x = x-\- x^ and y = y ^y^, the moment of 
 inertia about our axis is Sm (^2 _|- ^2) = 2m (^2 -|. yi) 
 + Sm {x'"^ -h ^'2) since ^mx^ = llmy^ = 0. Hence the 
 moment of inertia about any axis is equal to the moment 
 about a parallel axis through the centre of inertia plus 
 the moment of the whole mass concentrated into the 
 centre of inertia about our axis. The moment of inertia 
 about an axis through the centre of inertia is therefore 
 less than that about any other parallel axis. 
 
 10. Impulsive Forces 
 
 We have already seen that a force acting continuously is 
 measured by the mass it acts upon into the acceleration it 
 produces in the mass in unit time. We now wish to deter- 
 mine how a force acting only for a brief interval may be 
 measured. A force cannot of course act instantaneously or 
 
28 NATURAL PHILOSOPHY 
 
 for absolutely no time, for in such a case to produce 
 any finite effect the force would have to be infinite and 
 there is no such thing as an infinite force. 
 
 Two elastic balls of mass, wi and W2, and velocities 
 Vi and V2, meet, going either in the same or opposite 
 directions. We shall suppose that no energy is lost by the 
 impact or no heat developed. The velocity of the first ball 
 is changed, not instantaneously, but in an exceedingly 
 short interval of time, from vi to V]_\ and that of the second 
 ball from V2 to V2'. During the short interval that they are 
 in contact they must move with the same velocity, v, 
 and this velocity is the average velocity while the change 
 in the velocities is being effected. During the time of 
 contact the first ball has changed its velocity from vi to ^1' 
 and its average velocity during this time must have been 
 
 2 
 
 while the average velocity of the second ball was 
 
 1)2 -\- V2' 
 
 — X . The kinetic energy of the system remains un- 
 
 1 J XI ^ m\vi2 m2V2^ mxvi'^ m2V2^ 
 changed, so that -^ 1 ^^ = 2 ' 2 * 
 
 Hence mi r ^ 2 ^ ) ^^^ ~ "^^'^ "^^ \~Y^ ) ^^^' ~ '^^^' 
 But — ^ — = ^ — = V, and wi(^i — Vi) = m2\V2 — V2). 
 
 We have thus the measure of a force which acts for a 
 short time and which is called an impulsive force or an 
 impact. The measure is the increase (or decrease) of the 
 momentum which it produces in a body. 
 
 11. Pendulum 
 
 We have seen that the time of a complete small oscilla- 
 tion of a suspended particle is lir-J—. If we have a rigid 
 
 body of mass M oscillating about a horizontal axis, the 
 gravitational couple is Mgh sin t?, where h is the distance 
 
PENDULUM 29 
 
 of the centre of inertia from the axis and t? the angle it 
 makes with the vertical. If I is the moment of inertia 
 about the axis, !-& = —Mgh sin ^ (1). li k is the radius of 
 gyration about a parallel axis through the centre of inertia, 
 I = M{k2 -\- h^). For a small oscillation sin ^ is sensibly- 
 equal to z?. Integrating (1), I? = -J — ^ — V??i2 _ ^2, 
 where t9i is the maximum excursion. Integrating again, 
 t = ^^IjhJjL f y — sin Y I • The time of a complete 
 
 oscillation is T = lir -J "^ and the length of the 
 
 T gh 
 
 J^2 J- /j2 
 
 equivalent simple pendulum is I = 7 
 
 If we wish to find the time for any amplitude we may 
 proceed as follows: Integrating (1) ?? = 
 
 Jl V2(cost?-cos^o) = iJ^Jsin^^- sin2!L. 
 Let sin ^ sin <^ = sin y » where (p is an auxiliary angle. 
 When t? = 0, ^ = and when ^ = 1^0, <P = w-' Hence dt = 
 
 € 
 
 dip 
 
 ^l — sin2-^sin2<p 
 
 2 
 
 By the binomial theorem the radical can be developed into 
 1 ^ 13?? 
 
 1 + -^ sin2 -^ sin2^ + "2' 4" ^^^* Y ^^^^ ^ "^ ^^^' 
 
 TT 
 
 Now J' sin2» ^rf^ = l^i. -|. A . . .2!L^y 
 
 Hence the time of a complete swing is 
 
 r...Vf['4)"-^Gi)"''4' + -] 
 
 From this rapidly converging series we can determine the 
 time for any amplitude as closely as we please. 
 
30 
 
 NATURAL PHILOSOPHY 
 
 12. Centres of Oscillation and Percussion 
 
 An impulsive force acting upon a body obviously im- 
 parts to it an impulsive velocity of the centre of inertia, or 
 an impulsive momentum of the entire mass considered as 
 concentrated at this centre, together with an impulsive 
 moment of momentum about an axis through this centre. 
 If a body (Fig. 6) is struck a blow at B in the direction 
 AB, the translational motion of the centre of inertia, G, is 
 the same as if the entire mass were 
 concentrated there and the force acted 
 on it parallel to its direction, and the 
 rotational moment of momentum about 
 an axis through G is the same as if this 
 centre were fixed. The impulsive veloc- 
 ity, V, of the centre of inertia, parallel to 
 AB, will move it a distance vdt in the 
 first small interval of time, and the ro- 
 tation about an axis through G, JL to 
 the plane ABG, will turn the body 
 through an angle d-^ in this time. This is 
 equivalent to a rotation about a parallel axis through some 
 point in the line 06", ± to AB. Calling OP, /, and OG, h, 
 the moment of the blow F about G is measured by the 
 impulsive moment of momentum about the axis through 
 
 (7, or F{l-h) = Mk2 t? and ^ = ^^^ " ^^ 
 
 Mk^ 
 
 -, where k is the 
 
 radius of gyration about the axis through G. The blow F 
 is also measured by the impulsive momentum of the 
 
 centre of inertia, or F 
 k2 + h2 
 
 Mv = Mh& = Mh 
 
 F{1 - h) 
 Mk2 
 
 , and 
 
 / = 
 
 h 
 
 The axis of spontaneous rotation for the 
 
 first instant, or the instantaneous axis, is therefore at a 
 distance from the line of the blow equal to the length of the 
 equivalent simple pendulum. Relative to an axis through 
 0, the point P is the centre of oscillation, and it is also a 
 
HARMONIC MOTION 31 
 
 centre of percussion, for a blow applied at this distance 
 from the axis of spontaneous rotation would cause no 
 reaction upon it. If we strike a ball with a bat or a tree 
 with an axe at a distance from our hands equal to the 
 length of the equivalent simple pendulum, we experience 
 no "sting"; otherwise we do. The subsequent motion of 
 the body consists simply of the initial impulsive velocity 
 of the centre of inertia, and the impulsive rotation about 
 the axis through it, with the superposed effects of sub- 
 sequent forces. The trajectory of a struck ball is determined 
 by the initial velocity of the centre of inertia, the spin 
 not being influenced by gravity. 
 
 li GP = h\ I = h + h^ and hh^ = k^. Hence the points 
 and P are mutually centres of oscillation and percussion, 
 one for the other, and the times of oscillation (under 
 gravity) are the same for either suspension. This gives a 
 means of determining the length of the equivalent simple 
 pendulum. 
 
 13. Harmonic Motion 
 
 Distant bodies influence each other by gravitational 
 (longitudinal) waves through the ether. Since such forces 
 radiate equally in all directions from a central point, the 
 intensity of the force must fall off as the square of the 
 distance, or the simple geometrical law must hold. Since 
 two equal bodies radiate twice the energy of a single body, 
 the force must be proportional to the mass, for the same 
 distance. Again the action of such a force upon two equal 
 distant bodies must be twice the action upon a single body. 
 Hence we have almost axiomatically Newton's law of 
 attraction — "All bodies attract each other with a force 
 proportional to the product of their masses and inversely 
 
 as the square of the distance, or / = — r— • The action 
 
 is purely mutual, and the forces urging each body are pre- 
 cisely equal, and opposite. 
 
 Let us consider the mutual force between a homogeneous 
 
32 NATURAL PHILOSOPHY 
 
 Spherical shell and unit mass at any point. Let a be the 
 
 density of the matter on the shell, or mass per unit surface, 
 
 and h the distance of the point from the centre of the 
 
 shell, r its radius, p the distance of the point from any 
 
 element of the shell and ^ the angle between any radius 
 
 and h. From symmetry it is evident that the resultant 
 
 attraction of the shell must be along h. An elemental ring, 
 
 concentric with the axis h, has a mass l-Kar^ sin §d^. 
 
 The component along k of every such elemental ring is 
 
 27r(Tr2 sin ?? ,, .. ,. 
 {h — r cos z?) d^. 
 
 pdp 
 
 rh 
 
 Since p2 = h^ -{- r^ — 2rh cos ??, sin ^ d^ = 
 
 and h — r cos t? = x; 
 
 2h 
 
 The total attraction is 
 
 TTcrr fft + r (/, + ^) (k - r) , , ircrr r* + ^ , ,.. 
 
 ■^. - J ? — '^ ^ ^. - J '^^^^'^^;^ 
 
 There are three cases: li h > r, we have / = — t-—: 
 
 \i h = r, f = 4x0-: and if h < r, we have to take the 
 limits as r -\- h and r — h, and the integral becomes 
 
 Hence any homogeneous spherical shell attracts, and is 
 attracted by, any external mass precisely as if its mass 
 were concentrated at its centre, but within the shell it 
 exercises no attraction, or there is no force. And any 
 sphere made up of homogeneous spherical shells attracts in 
 the same way. 
 
 Let us suppose a homogeneous sphere with two diame- 
 ters bored out _L to each other. The attraction of the 
 
 sphere on unit mass at the surface is -^r = -^ — , and the 
 
 attraction at any level in the interior is ^ , where r is 
 the distance from the centre. Hence if we drop unit mass 
 
HARMONIC MOTION 33 
 
 into one of these holes, it will oscillate harmonically be- 
 tween two diametrically opposite points of the surface, the 
 motion being harmonic because the force is proportional 
 
 to the distance from the centre. Let 2^-^ hek. If we pro- 
 ject the mass along the surface with a velocity, v = kR, 
 which makes the centrifugal force just equal to the 
 attraction, it will revolve about the sphere, just grazing 
 the surface. The time for the outside mass to traverse a 
 
 quadrant is -^ ,and the time to reach the centre is the same. 
 
 1 
 
 k2r, r = k^ R2 — r^ and ^ = rsin -^ 
 
 R 
 
 Hence the outside and inside masses will regularly meet as 
 the outside mass passes over each hole. Using the holes as 
 axes of X and y, the positions of the inside masses will be 
 the co-ordinates of the outside mass. 
 
 The inside bodies execute simple harmonic motions, 
 while the outside body executes a compound harmonic 
 motion made up of two equal simple harmonic motions ± 
 
 1 -If 
 to each other. Since ^ = ^- sm — 
 
 k To 
 
 where Yq is some 
 
 level within the sphere from which we drop a body, the body 
 
 thus dropped will have the same period as if dropped from 
 
 the surface. Thus the period of any harmonic motion 
 
 is independent of the amplitude of the motion. It is for this 
 
 reason that such a motion is called harmonic. The vibrating 
 
 parts of all musical instruments execute harmonic motions. 
 
 Otherwise the period (pitch) would change with the 
 
 intensity (amplitude) and music or harmony would become 
 
 impossible. 
 
 Two ± simple harmonic motions with the same period 
 
 and amplitude, and one of the motions a quadrant in 
 
 advance (or behind) of the other, result in a circular 
 
 harmonic motion. If the amplitudes are not equal we have 
 
 TT 1 . ~^^ 
 
 elliptic harmonic motion. Since ^ = -^ + -r- sin ^^ for 
 
34 
 
 NATURAL PHILOSOPHY 
 
 one mass when ^ = r sin - for the other, we may write 
 
 y = ro sin kt and 
 
 R sin 
 
 X2 
 
 (" - r) 
 
 R cos kt. 
 
 R2 
 
 + 21 = 
 
 1, or the path for the com- 
 
 pound motion is an ellipse. The angle in these expressions 
 is called the phase of the motion and the difference be- 
 tween the phases of the two components is always y- 
 
 The reciprocal of the period is called the frequency, or the 
 
 number of vibrations in a second. The complete period is 
 
 27r 
 
 — . It is evident that any number of simple harmonic 
 
 motions making any angles with each other, but all having 
 a common centre, can be compounded. If there is a com- 
 mon period the result will be a steady elliptic motion. 
 If the periods are different the resultant motion will be 
 continually shifting, forming what are known as Lissajon's 
 curves. If the periods have a common multiple the changes 
 will periodically repeat themselves. 
 
 14. Tidal Forces 
 
 A homogeneous ring revolves about an attractional 
 centre, S, which is in its plane, and the plane of the ring 
 
 is ± to its orbit. The 
 distance SC = D is 
 constant and (p is the 
 angle any element of 
 the ring makes with 
 CA . D is so great that 
 all lines from 5 to the 
 ring may be regarded 
 as sensibly parallel. 
 The centrifugal force 
 for any element is (D + r cos cp) yj/^rdip, where yp is the 
 orbital angular velocity. Integrating, we find that the 
 
 Fig. 7. 
 
TIDAL FORCES 35 
 
 centrifugal force for the outer half is TrrDxp'^ + 2r2^2^ and 
 for the inner half 7rrD\l/^ — 2r2^2. Calling the mass of the 
 ring, M, these two centrifugal forces are 
 
 -^ Z)i/'2 ^ —-. — p and -^ DrP^ " T' — "l"^' 
 
 Ir 
 
 — is the distance of the centre of inertia of a half ring from 
 
 C, so that the rotational centrifugal force for each half of 
 the ring is the same as if its mass were concentrated at its 
 centre of inertia. 
 
 Let / be the acceleration due to the central attraction 
 at unit distance. Then the total attraction of the ring is 
 
 J frdip r2/ sin (p _ 
 (D + r cos <py' ~ (D2 _ r2) {D -\- r cos ip) 
 
 -r=- — .^ cos "F— -^ • Taking the limits for the 
 
 (£)2 — r2)| D + r cos ^ ^ 
 
 outer half, and considering that {D^ — r^) is nearly equal 
 
 — ^T IT 
 
 to D2 and the angle cos j^ is nearly equal to -x, we have 
 for the attraction on the outer half — -^ H — ^rr-- Like- 
 wise the attraction on the inner half is — r=^ ^- If the 
 
 L/2 JJ^ 
 
 total centrifugal force, which is MDyj/^, balances the total 
 
 Mf 
 attractional force, which is — jr^ , all the forces will be in 
 
 equilibrium, and the motion will be equivalent to a mass 
 concentrated at C revolving about 5 at a constant distance 
 with constant angular velocity. But we have a force acting 
 
 on the outer half, ( ^ + i/'^ ), which is balanced by an 
 
 equal and opposite force acting on the inner half, 
 
 (-^ H-i/'^V These two forces, therefore, tend to 
 
 IT \V'^ f 
 
 pull the ring apart in either direction from the centre, the 
 inner half being urged towards the attracting body, while 
 
36 NATURAL PHILOSOPHY 
 
 the outer half is urged directly away from it. If the ring is 
 
 not perfectly rigid, and perfectly rigid bodies do not exist, 
 
 it will be deformed into an oval pointing towards the 
 
 Mr f 
 attracting body. The differential forces jr-^ and 
 
 ^ are called Tidal Forces. As a planet can be con- 
 
 sidered to be made up of rings which are concentric on a 
 line Jl to the plane of our ring, it is evident that these 
 tidal forces are continually deforming it into an oval 
 which always points towards the attracting body. The 
 moon and the sun both deform the earth into ovals point- 
 ing towards them. The longest axis of the oval is towards 
 the attracting body, while the shortest axis is ± to the 
 orbit. 
 
 The effect upon the rotation is similar to a friction 
 band tending to slow the rotation down to coincidence 
 with the revolutional period, and eventually this must 
 occur. The rotation cannot fall below the revolution for in 
 that case the tidal band would accelerate it into coinci- 
 dence again. The effect in the case of the earth is very 
 slight but the height of this solid tide may possibly approxi- 
 mate a foot. We do not know exactly what it is. It is 
 constantly at work and it has very important and far- 
 reaching effects which we shall discuss later. The earth is 
 always rotating about two axes — one being the diurnal or 
 polar axis, the other being the precessional axis which is ± 
 to the former. The effect of the tidal brake-band is to 
 oppose these two rotations, with the result in one case of 
 lengthening the day and in the other of changing the inclina- 
 tion of the earth's axis. The oceans constitute a thin 
 skin of water covering about ^ of the earth's surface and 
 they are of course subject to the tidal action we have just 
 discussed. Theoretically, the effect of these ocean tides is 
 to decrease the earth's motion, but compared with the 
 solid tides of the earth's mass the effect is insignificant. 
 Thus while the solid tides will undoubtedly bring the 
 
ATTRACTIONAL AND MOLECULAR RIGIDITY 37 
 
 rotation and revolution into coincidence in some finite 
 time, the ocean tides could only effect this in anin- 
 finite time. As our earth has not, and will not exist 
 for an infinite time, such ocean effects are wholly negli- 
 gible. 
 
 When two bodies are near each other, it will be seen 
 
 that the terms containing -^ and xj/^ become very large, 
 
 and the rending force may reach a limit beyond which the 
 two bodies would be torn apart. Since these forces are 
 also proportional to r, this limit would be reached sooner 
 in larger bodies. 
 
 The particles of an attracted body, instead of executing 
 circles about the rotation axis, are forced to move in 
 ovals and as their distances from each other are now 
 greater and now less, they are continually kneaded, 
 stretched and compressed, and heat is produced, just as a 
 hammered bar becomes hot. All this heat energy which is 
 dissipated is produced at the expense of the rotation. 
 When the rotational and revolutional periods coincide 
 this action ceases. Such a coincidence of periods probably 
 exists in all the satellites of our system, and in Venus and 
 Mercury. Our moon is a near example. 
 
 15. Attractional and Molecular Rigidity 
 
 In a very large mass, such as the earth, there is no in- 
 herent elasticity of figure such as exists in small masses and 
 which is due to the interaction of adjacent molecules. 
 A small mass can have any shape while a large mass, of 
 itself, has only one shape, viz., a sphere. It would be im- 
 possible for a body as large as the earth or the moon to 
 have the shape of a bar. We must thus distinguish between 
 two kinds of elasticity, or force tending to restore a body 
 to its original shape after deformation. 
 
 Our conceptions of matter are formed from the con- 
 ditions under which it exists at the earth's surface, but it 
 is unsafe to project these ideas into places where the 
 
38 NATURAL PHILOSOPHY 
 
 conditions differ widely. A solid spring regains its shape 
 because of the cohesion between its adjacent molecules and 
 it possesses only a proximate molecular rigidity. As the 
 size increases this kind of rigidity becomes of less im- 
 portance until it is entirely overcome by self -gravitation 
 and the mass flows into a sphere. A fluid is etymologically 
 that which flows, so that in saying that the mass flows into 
 a sphere we imply that it is no longer a solid body but 
 a fluid body, although at the surface small portions may 
 still be regarded as solid. When we melt a solid it loses 
 largely its molecular rigidity and assumes a form of equi- 
 librium depending chiefly upon gravitational forces. The 
 matter in the interior of a large body, such as the earth, is 
 thus certainly fluid or plastic because the shape it assumes 
 is due solely to the enormous gravitational pressures and 
 not at all to any molecular rigidity. The question as to 
 whether the interior of the earth is solid or liquid is 
 founded upon a confusion between general attractional 
 rigidity and molecular rigidity. It is certainly plastic and 
 very hot — probably well above 4000°C from the centre 
 to a few miles from the surface. For all practical pur- 
 poses, it is therefore in a molten condition. 
 
 A very large body must necessarily have a very high 
 rigidity, exceeding that of a very small body, but the 
 resistance to deformation is of a very different nature in 
 the two cases and it is difficult to compare them. The 
 earth has a high rigidity but it is manifestly improper to 
 say that it has the rigidity of a globe of steel or of a globe 
 of glass of equal size. 
 
 We may determine the rigidity of a glass marble, which 
 is the rigidity of its proximate molecules, but if we in- 
 creased it to the size of the earth, this particular kind of 
 rigidity would become insignificant in comparison with its 
 general attractional rigidity. A small piece of glass might 
 have any shape, but a large piece could have only one 
 shape. A further difficulty occurs in that a globe the size 
 of the earth could not exist as glass or steel throughout. 
 
DENSITY OF THE EARTH 39 
 
 The enormous condensation towards the centre would 
 necessarily result in changing the kind of matter. We have 
 previously referred to a suggestion that the ether may be 
 the ground stuff out of which all the "elements" have been 
 formed under varying conditions of pressure and tempera- 
 ture. This is merely a surmise, though perhaps a natural 
 one when we see denser elements spontaneously splitting 
 into lighter ones. We know that the average density of 
 the earth is 5^ times that of water, while the surface 
 density is only about 2^. There must be much matter in 
 the interior which is much denser than the surface matter 
 and it seems probable that such matter may be like some 
 of the denser elements with which we are familiar. A 
 considerable part of the earth's interior is probably iron. 
 Magnetic phenomena point to much paramagnetic matter 
   — iron, nickel, cobalt. Siderites seem to be fragments of 
 some former body, and their usual iron-nickel composition 
 suggests that many bodies in the universe are automatic- 
 ally largely composed of these substances. 
 
 16. Density of the Earth 
 
 The matter of the earth was probably once homogeneous, 
 but by self -gravitation has condensed to its present state. 
 The density of this matter must, on the whole, increase 
 regularly and progressively from the surface to the centre. 
 What the exact law is, we do not know. Laplace assumed, 
 as a hypothesis, that the increase of the square of the 
 density was proportional to the increase of the pressure. 
 That is to say the compression gradually decreases as the 
 density increases. While perhaps not the actual law, it 
 would seem that Laplace's formula must represent the 
 actual conditions to a close approximation, even bearing 
 in mind the possibility that the condensation may at 
 certain stages have been per saltum, as where an element 
 may have changed suddenly into a denser one. 
 
 Let r be the distance of any level from the centre, p 
 and / the density and acceleration at that level, and pa the 
 
40 NATURAL PHILOSOPHY 
 
 average density of all the matter within that level. Taking 
 the direction outward as positive, 
 
 ■^ "" ~ r2 I ^'^f^^^^^ = ~ -T ^Pa- 
 
 3 
 
 Hence rDrPa = 3 {p — Pa) (2). 
 
 Laplace's formula is pdp = kdp, where p is the pressure 
 and k is some constant, dp = pf.dr. Whence DrP = kf (3). 
 p is a function of r, or for any value of r there is a single 
 definite value for p. 
 
 p = ^ (r). By Maclaurin's theorem 
 
 P = <P (o) -}- <pi (o) r + <P^^ (o) 2j + etc. 
 
 DrP = <p^{r) = kf. <p^^ (r) = kDrf = ^rk 
 
 (j'^-') 
 
 2 (P - Pa) _^- 
 
 ^iii(r) = ^Trk \j DrPa - kf\ =47r^r 
 
 ,Kr)=-[Mfl(l)+,uw+i^]. 
 
 (p{o) = pc, where pc is the density at the centre. 
 <p^{o) = 0, since the force at the centre is zero. 
 
 4:Trk 
 
 <p^^ (o) = :T-pc- v^^K^) becomes indeterminate when 
 
 r is zero, but by differentiating numerator and denomi- 
 nator of the indeterminate term, we obtain zero as the 
 limiting value. 
 
 Limit (piv{r) = — Sirkip^^ir) — 4c(piv(r), 
 {r = 6>) (r = o) {r = o) 
 
 35 7r2yfe2 
 and limit <piv{r) = — t-e — Pc- Similarly we find that ^»(o) 
 
 (r = 0) 
 — 0. All the odd orders of <p{o) are zero and all the even 
 orders are multiples of pc with increasing powers of k 
 in the coefficient. Thus (pvi{o) is a multiple of — ^^pc,et c. 
 ^ is a very small quantity so that we can neglect higher 
 
DENSITY OF THE EARTH 
 
 powers than the first. Hence we have 
 Denoting the average density of the body by pa, 
 
 41 
 
 (4) 
 
 
 
 and 
 
 k = 
 
 (-» 
 
 (5). 
 
 and 
 
 IttRP- 
 
 Hence, 5pa = 2pc + Sp^ (7), where Ps is the surface density. 
 We have then the general law that for any large body 
 (sphere), five times the average density is equal to twice 
 the central density plus three times the surface density. 
 In the case of the earth, many surface rocks have a density 
 around 2.75. Assuming this as the average surface density, 
 which is about half the average density of the whole earth, 
 
 we have Pc == y ^^' 
 
 or the density at the centre of the earth 
 
 is 3>^ times the surface density, or it is 9.63 times the 
 density of water. 
 
 From Equa. (1) it will be seen that if the surface 
 density of any body is less than ^ of its whole average 
 density, there will 
 be a level between 
 the surface and the 
 centre where fzpa^ 
 p, and DrJ = 0. At 
 this point / is a 
 maximum and the 
 p curve has a point 
 of inflexion. At this 
 particular level the 
 
 body exerts a greater attraction than at any other distance 
 from the centre, and we shall call it the critical level. In 
 the earth diagram, Fig. 8, this level is shown at A . 
 
42 NATURAL PHILOSOPHY 
 
 Let X = AS = R — r, where r = CA. 
 
 5 r2 
 R (Pa - 2ps) 
 
 2 {Pa - Ps) 
 
 and 
 
 Pa = Pc 
 
 3 R2 ^ 
 
 Combin- 
 
 5_ r2_ _2_ 
 3 R2 3 
 
 ing these equations we find that r = .94i?. Or the critical 
 level of the earth lies about 240 miles below the surface. 
 The attraction therefore increases gradually up to this 
 point after which it decreases regularly to zero at the 
 centre. Only bodies of a certain size can have a critical 
 level and the fact that this level is so near the earth's 
 surface would seem to indicate that bodies of a little less 
 size would have none. The moon probably has no critical 
 level, or its attraction is a maximum at its surface, 
 decreasing both towards and away from the centre. 
 
 Taking as our units, feet and seconds, the unit of mass 
 will be that mass which attracts a like mass at unit dis- 
 tance with unit force, or a force giving it unit acceleration 
 in unit time. The earth attracts unit mass at its surface 
 with a force of 32 units. It would therefore attract unit 
 mass at unit distance with 32i^2 units of force, and unit 
 
 M 
 mass IS -ry^, where M is the mass of the earth. The mass 
 
 47r 
 of the earth is equivalent -r- ^^ X ^-^ c^- ^^- of water. Hence 
 
 unit mass is equivalent to .72R cu. ft. of water, where R is 
 
 expressed in feet. Or the matter of some 14 million cu. ft. 
 
 of water concentrated into a point would attract a like 
 
 mass at the distance of a foot with a force which would give 
 
 it a velocity of a foot a second for every second that it acted. 
 
 Unit density is this unit mass concentrated into a cubic 
 
 foot. Hence, expressed in these units, the average density 
 
 7 68 
 of the earth is-^ = 3.68 x 10"^, and the surface density 
 K 
 
 • 3.84 ^ ^, ^ . / , 
 
 IS — ^ . In the same way we find that the pressure at the cen- 
 tre of the earth is about 49 million pounds per square inch. 
 
GREEN'S THEOREM 43 
 
 We shall refer here briefly to certain arguments for the 
 solidity of the earth's interior. With the feeble pressures 
 available in our laboratories it has been found that bodies 
 which expand on melting have their melting points raised 
 by pressure while bodies which contract on melting have 
 their melting points lowered by pressure. It is argued that 
 the enormous pressures in the interior will prevent the 
 matter from liquefying. Further, certain surface rocks 
 have been melted and it has been found experimentally 
 that there is a slight expansion. On the other hand the 
 molten lava lake of Kilauea is often skimmed over with a 
 solid crust just as a lake of water is covered with a sheet 
 of ice in winter, and in both cases the crusts are readily 
 supported by the underlying liquid. This would indicate 
 that the deeper matter, from which the lava comes, con- 
 tracts on melting. Again iron contracts on melting and 
 there is little doubt that iron forms a considerable part of 
 the earth's interior. According to the argument, therefore, 
 a very considerable part of the interior matter may have 
 its melting point lowered by the enormous pressures and 
 is therefore liquid. But such arguments are entirely 
 inapplicable. Beyond a certain limit of pressure all matter 
 becomes fluid, or flowing, and plastic. The interior of the 
 earth is certainly plastic and very hot, and is therefore 
 to all intents and purposes in a molten condition. The 
 extrusion of molten rock from all parts of the earth's 
 surface points strongly to such a general condition. Equally 
 futile is the argument that the earth must be solid to have 
 its high rigidity. The earth has a very high gravitational 
 rigidity, but no molecular rigidity except at the surface. 
 Surface conditions of matter cannot be projected into 
 the interior. 
 
 17. Green's Theorem 
 
 Let us suppose that within a closed surface, 5, 1^ is a 
 function which has a single finite value for every point of 
 our enclosed space, and varies continuously (without 
 
44 NATURAL PHILOSOPHY 
 
 abrupt change) in any direction. Choosing any axes, a 
 line parallel to the x axis must cut such a surface an even 
 number of times. 
 
 s 
 
 . dx = W2 — Wi, where W\ is the value of W at the 
 
 point of entrance and W2 its value at the point of exit. 
 
 I I I ~a~ ' ^^^y^^ ^ I I Wdydz, the volume integral on 
 
 the left being taken throughout the whole closed space, and 
 the surface integral over the whole surface. Representing 
 the volume element, dxdydz, by dr and the surface element 
 dydz by dS cos (nx) , where dS is the element of the closed 
 surface cut out by an elementary parallelopiped and 
 (nx) is the angle between x and the normal to the surface, 
 
 drawn inward, j I 1 -^ dr = 1 j W co^ {nx) dS . W is 2iny 
 
 continuous finite point function, and if U -r- be such a 
 
 dx 
 
 function, we can substitute it for I^, or j I j — ( U — \dT 
 
 = I I L^ -T— cos {nx) dS (1), with similar expressions for 
 y and z. Adding, 
 
 (fcm^ + ^^ + ^^y. 
 
 J J J \ dx dx dy dy dz dz J 
 
 ""III "^ — ^^^ ^^^^ "^ ~^ ^^^ ^^^^ "^ ~^ ^^^ ^'^^^ ) 
 
 This result is known as Green's Theorem. The derivatives 
 here are partial as indicated by the notation and obviously 
 
 — — cos (nx) + ^-- cos {ny) + -r— cos {nz) = -7 — , where 
 dx dy dz dn 
 
 dV 
 
 -r — signifies differentiation in the direction of the normal. 
 dn 
 
GAUSS' THEOREM 45 
 
 Hence we can write (2) 
 
 dU dV dU dV dU dV 
 dx dx dy dy dz dz 
 
 Since, by symmetry, U and V are interchangeable, we have 
 
 r r c[vf—-\-—4-—\- u{—4-—-^-—\ 
 
 J J J L v^^ ^y^ ^^v v^^ ^y^ ^^v 
 
 (3). Equa. (3) is known as Green's Theorem in its second, 
 form. 
 
 18. Gauss* Theorem 
 
 Since every atom is always in motion it radiates energy 
 incessantly by communicating its motion to the ether. It is 
 thus the centre of a field of force and the lines of force are 
 straight lines radiating from this centre, while equipo- 
 tential surfaces are spherical surfaces. The force at any 
 
 point is , where m is the mass of the radiating point. 
 
 The integral of this with respect to r, in the direction of the 
 force, is the work done, and the integral in the opposite 
 direction is the work undone which is called the potential, 
 or stored, energy, represented by V. Therefore V = —W. 
 If we assume that attraction is effected through the agency 
 of longitudinal waves in the ether, it can be shown (Me- 
 chanics of Electricity) that the force at any point is equal 
 to the quantity of wave energy traversing (orthogonally) 
 unit surface in unit time. In other words, the force is 
 measured by the density of the flux of the energy at the 
 point. The total energy traversing any surface surrounding 
 the point is necessarily the same as that crossing any other 
 such surface, or is equal to the flux of energy across unit 
 sphere about the point. The density of the energy at any 
 point is necessarily inversely as the surface, or as the square 
 of the distance, so that our assumption that attraction is 
 
46 NATURAL PHILOSOPHY 
 
 effected by longitudinal waves contains implicitly the 
 
 result,/ = . The total flux of energy across any spherical 
 
 surface is 2/ = 47rm, and this is the energy flux across any 
 surface surrounding the point. It follows that for any 
 number of points, or for any distribution of matter, the 
 total flux across any surrounding surface is ^irM. If the 
 surface does not surround the radiating matter, since the 
 energy cannot accumulate within this surface, but the 
 amount of energy contained by the surface must at all 
 times be constant, it follows that as much energy is always 
 passing out of the surface as is entering it, so that the total 
 flux of energy across its surface is zero. These results 
 constitute Gauss' Theorem. Mathematically expressed, 
 
 2 4^ c/5 = -S -^ cos {nr) dS = 0, 
 an fi ^ ' 
 
 when the matter is without the surface, and 
 
 2 4^ J5 = -2 4- cos (nr) dS = 47rM, 
 dn 7-2 ^ ^ 
 
 when the matter is within the surface. The normal, 
 
 w, is always supposed drawn inward from the surface. 
 
 The potential due to a particle, w, at any point is the 
 
 fyi 
 scalar (undirected) quantity, F = — , and the potential 
 
 T 
 
 at that point due to any number of particles is 2 — • 
 
 19. Poisson's Equation 
 
 Selecting a certain point, let us surround it with a sur- 
 face. This surface may cut matter, so that there is matter 
 both within and without the surface. The total flux of 
 energy through the surface from the outside matter is zero, 
 while that from the inside matter is 
 
 r [^"^^ ^ ~ f [^'"''^ ^'^^^ dS=47rM=^T r r fpdr 
 (1), p being the density of the inside matter at any point. 
 
ATTRACTION OF A CIRCULAR DISC 47 
 
 From Green's theorem, putting U = 1, 
 
 //^-=-///(£-r-g-r.|?)-<«. 
 
 Designating the operation, (^ + ^ + ^)t>yA, and 
 
 combining (2) with (1), we have 
 
 j j j (aV -\- 47rpj dT = (3). If we allow our surface to 
 
 close down upon our point, Equa. (3) will still be true. 
 Therefore, AV -\- 47rp = 0, which is Poisson's equation. 
 It expresses the fact that if the potential at any point due 
 to any distribution of matter, be operated upon by the 
 operator A, the result will be — 47rp, where p is the density 
 of the matter at that point. If there is no matter at the 
 point, AF = 0, which is Laplace's equation. 
 
 unit mass situated in its axis, is I TT", — ZTJ dr, where R 
 
 IS 
 
 20. Attraction of a Circular Disc 
 
 The attraction of a homogeneous circular disc upon 
 
 lircrar 
 
 (r^ + a2)i 
 
 is the radius of the disc, a the distance from the disc, and <r 
 the density of the matter upon it. Integrating, 
 
 F = lira- I 1 =^= I = lira (1 — COS a), where a 
 
 \ ^R2 - a2/ 
 
 the angle subtended by R at the attracted point. If the point 
 is in the surface, F = lira. In crossing the surface, since the 
 attraction is lira on one side and — 2x0- on the other side, 
 there is a sudden change, or discontinuity, in the value of 
 F of 4^(7 . And in traversing any shell, no matter how mat- 
 ter is distributed over it, the change of the attraction must 
 always be 4x0-. For the attraction of an infinitely small 
 disc at the point will be 2x0-, and that of the rest of the 
 shell could not have changed during the infinitely small 
 motion of traversing the shell. We have already seen that 
 the attraction of a spherical shell (homogeneous) changes 
 from 47ro- just outside the shell to zero within the shell. 
 
48 NATURAL PHILOSOPHY 
 
 The attraction of an infinitely small disc of the shell at 
 the point was lira. Hence the attraction of the rest of the 
 shell was also lira. In traversing the shell the latter did not 
 change, but the former changed to —Ittct. Hence the 
 attraction within the shell became zero. 
 
 21. Maclaurin's Theorem 
 
 We have an ellipsoid, -^ + fr + T = 1» uniformly 
 
 filled with matter of a density, 7~"("7-t-r7 + ~)» and 
 
 we wish to find if it is possible to distribute negative matter, 
 or matter which repels instead of attracting, over its 
 surface, in such a way that it will annul the external 
 attractive field of the ellipsoid. For the purposes of analysis 
 it would suffice to make use of an imaginary substance 
 which we supposed to repel instead of attracting. Such a 
 result would be obtained by simply making the density of 
 such a substance negative instead of positive, as by making 
 such changes in our formulas we reverse attractional 
 actions. It is certain that these actions are effected through 
 and by the ether, and probably by the agency of longitu- 
 dinal waves. Assuming this agency, and calling matter 
 denser than the ether, positive matter, and matter less 
 dense than the ether, negative matter, it can be shown 
 (Mechanics of Electricity) that positive matter attracts 
 positive matter, while it repels negative matter. The 
 repulsions of nature, of which there are many instances 
 (comets' tails), as well as attractions, are probably to be 
 explained in this way, and in all these cases the actual 
 motions are accurately expressed analytically by simply 
 making the density negative. The negative matter which 
 we postulate in the present problem is therefore not merely 
 an imaginary or mathematical conception, but probably 
 represents an actual condition in nature. 
 
 But to return to our problem. If such a distribution of 
 negative matter on the surface of the ellipsoid is possible, 
 
MACLAURIN'S THEOREM 49 
 
 then there will be no flux of energy through this surface, 
 and the potential at the surface and at all external points 
 must be zero. Gauss' theorem requires that the positive 
 and negative matter within the surface shall be equal, or 
 that the algebraical sum of this matter shall be zero. The 
 potential at any point, x, y, 0, within the surface must be, 
 
 1 / X2 y2 02 \ 
 
 V = y(l 2~h2 2")'^^^ ^^^^ makes the potential 
 
 zero at the surface, and at any interior point p = — 27- AF, 
 
 which satisfies Poisson's equation. 
 
 The distribution of matter over a surface, without any 
 thickness, like the concentration of matter in a point, is a 
 purely imaginary mathematical conception, but if we have 
 a shell bounded by two surfaces which though infinitely 
 near give the shell varying thicknesses at different points, 
 and we fill this shell with homogeneous matter of any 
 density we please, then we have a real distribution. 
 
 Since the surface of the ellipsoid is an equipotential 
 surface, the force inward from this surface must be every- 
 where normal, and this force is 
 
 f:-[(S)"+(¥)"+(f)"]'=[i+s+?i]' 
 
 Since in crossing the surface the force must change ab- 
 ruptly by an amount 47r(7, where a is the density of the 
 surface matter at the point of crossing, and since the 
 length, p, of the perpendicular from the centre to the 
 tangent plane at any point is 
 
 r^2 'y2 22-|-i 
 
 [^4 + 6"4 + ^J 'W^have 
 
 dV , 1 
 
 on p 
 
 Hence if we distribute negative matter over the surface 
 
 in such a way that at every point its density is — j- • -, 
 
 47r p 
 
 the problem is solved. 
 
 The equation of an ellipsoid in terms of the ±, p, 
 from the centre to a tangent plane at any point, and 
 
,2 
 
 50 NATURAL PHILOSOPHY 
 
 a, jS, 7, the direction angles of this ± referred to the 
 principal axes, a, b, c, is p^ = a^ cos2 a + 62 cos2 j3 -}- 
 c2 cos 27. For projecting the co-ordinates of the point on 
 the ± , we have p = x cos a -\- y cos ^ -\- z cos 7, and the 
 
 equation -:; + fr + — = 1. Varying only x and y in these 
 
 Cl^ u^ C^ 
 
 equations we have dx cos a -\- dy cos i3 = 0, and 
 
 — :- + ^-rr = 0. Hence -x cos /3 = ,-3^ cos a with similar 
 o2 o2 a 
 
 expressions for x and z, and y and s. ^2 = -j£;2 cos2 a + 
 
 );2 cos2 /8 + 22 cos2 7 + 2:J[:;y cos a cos /3 -f 2xz cos a cos 7 + 
 
 2yz cos /3 cos 7 = a2 cos2 a -\- b^ cos2 /3 + ^^^ cos2 7 — 
 
 /b ^ a \2 /c a \2 
 
 I -:j; cos p — -j-y cos o: J — ( -^ cos 7 z cos a j 
 
 — ( ry COS 7 2 cos j3 j 
 
 The last three terms are zero. 
 
 If we vary the lengths of the axes of an ellipsoid but 
 keep the direction of the ± to a tangent plane unaltered, 
 the variation of the length of p will measure the thickness 
 at the point of tangency of the shell formed by the original 
 surface and the varied surface. 
 
 cos2 a • 6a2 + cos2 ^ 8b^ + cos2 y dc^ 
 
 sp = 2^ 
 
 In order that the thickness of the shell, or amount of 
 
 matter in it at every point, shall be proportional to -» 
 
 P 
 we must vary the surface so that 5a2 = 562 = 5^2. 
 
 But this is a property of confocal ellipsoids, for the 
 condition of confocality is a2 — 62 = Const. 62 _ ^2 = 
 Const., or 5a2 = 562 = 3^2. 
 
 Calling the thickness of the shell at any point, t, and 
 
 the common variation of the squares of the semi-axes, q, 
 
 . 1 2t 
 
 we have — o- = -; 
 
 47r q 
 
 If then we have an infinitely thin shell, bounded by two 
 
 confocal ellipsoids, filled with negative matter, and the 
 
THEOREM ON ATTRACTION 51 
 
 inner ellipsoid filled with an equal amount of positive 
 matter, both matters being homogeneous, there will be no 
 external field. The field of the confocal shell is therefore 
 the same, but reversed, as that of a homogeneous confocal 
 ellipsoid of the same mass, and if the masses are not the 
 same the contours of the two fields, or lines of force, are 
 the same, but the intensities of the forces at different 
 points are proportional to the masses. 
 
 By increasing or decreasing our original ellipsoid through 
 the addition or subtraction of infinitesimal confocal shells, 
 but keeping always the same amount of homogeneous 
 matter filling the ellipsoid, we shall not alter the external 
 field. Hence any confocal shell has the same external field 
 as any other confocal shell with the same foci and mass, 
 or as any confocal homogeneous ellipsoid of the same mass. 
 If the masses are not equal, the direction of the force at all 
 external points is the same, but the intensity is proportional 
 to the mass. This is Maclaurin's Theorem. Generally, 
 all thick or thin homogeneous shells bounded by confocal 
 ellipsoids have the same external fields if their masses are 
 equal: if not, the forces at any point are in the same 
 direction and proportional to the masses. 
 
 Since in the limiting case of confocal ellipsoids, the 
 ellipsoid becomes an elliptic disc with semi-axes Va2 —c^ 
 and V62 — c2, an elliptic disc will attract the same as any 
 confocal ellipsoid, if we distribute uniformly upon the disc 
 an equal amount of matter. And a circular disc will exert 
 the same attraction on all external points as an ellipsoid 
 of revolution of equal mass, provided the radius of the 
 disc is Va2 — 62 and it is similarly placed. 
 
 22. Theorem on Attraction 
 
 The resultant attraction of a circular disc on a point in 
 its axis we have seen to be 2x0- (1 — cos a), (Art. 20). 
 27r(l — cos a) is the solid angle subtended by the disc at 
 the point. For if r is the radius of the sphere which con- 
 
52 NATURAL PHILOSOPHY 
 
 tains the circumference of the disc and has the point for 
 its centre, the surface of the spherical cap cut out by the 
 disc is 27rr.r(l — cos a). The attraction of the disc is 
 therefore measured by the solid angle, or surface of unit 
 sphere cut out by the cone formed by the disc. Generally 
 the resultant attraction of a homogeneous disc of any 
 shape in the direction OP, the _L from the point to the 
 disc, is measured by the surface cut 
 out of unit sphere, drawn about the 
 point, by the cone having the disc 
 as a base and the point as an apex. 
 An elementary cone do) cuts out of 
 a spherical surface through A, an 
 
 TO 
 
 element dS, and an element out 
 
 cos a 
 
 of the disc. The resolved part of the 
 
 attraction of the element in the di- ^^^- ^• 
 
 rection OP is — -, p being the distance of the element. 
 
 p2 
 
 S -— is the solid angle subtended by the disc at 0, and the 
 
 p2 
 
 theorem is proved. When the disc is symmetrical about 
 the ±, this measures the total resultant attraction. 
 The theorem gives a means of finding the attraction of an 
 ellipsoid on an external point, in the direction ± to its 
 focal, or a, b, plane. For we can reduce the ellipsoid to an 
 elliptic disc confocal with the ellipsoid and of equal mass, 
 and the attraction of this disc, ± to its plane, is the sur- 
 face cut out of unit sphere about the point by the cone of 
 the disc, into its density. 
 
 23. Homoeoids 
 
 For simplicity, we shall define a homoeoid as an infinites- 
 imal shell bounded by two similar ellipsoids. The con- 
 dition of similarity between two ellipsoids is 
 
 ^^ I y^ , 2^ _ . 
 
 a2 H+q)'^ b2 (1 + 5) "*" c2 (l-f g) " '' 
 
HOMOEOIDS 53 
 
 and we have seen that the condition of confocality is 
 
 X^ a;2 22 
 
 + 7-r^. — H — r—. — = 1. That is, the variation of 
 
 a2, 62^ ^2^ is qai etc., in the first case and q in the second. 
 
 ^ = V a2cos2a + 62cos2j3 + c2cos27, and the thickness, t, of 
 
 a shell at any point is the variation of the constantly 
 
 directed ± , />, to a tangent plane at that point, as we vary 
 
 the axes infinitesimally. 
 
 cos2a5a2 + cos2i3562 4- cos275c2 ^^ . ... ., 
 
 bp = t -=   TT- ' — . If the ellipsoids 
 
 ip 
 
 are similar, bp = ^ = ^ ; if they are confocal, bp = t = ~ . 
 
 Hence the density of matter at any point of a homoeoid 
 varies directly as p, while in a confocal shell it varies 
 inversely as /?. 
 
 Let us describe an infinitesimal double cone, Jco, having 
 some point, P, in the interior of a homoeoid as its common 
 apex. These cones will cut out elements, dS.dp, attracting 
 
 the point with a force — ir~^» where r is the distance of P 
 
 from the element, p = r sin r, where r is the angle between 
 r and the tangent plane at dS. dp = sin rdr, and the 
 
 attraction of each element is :; = drdoj, since 
 
 r2 
 
 = dco. Now any plane section of a homoeoid 
 
 must be two similar ellipses, and passing a plane through 
 the axis of our cones, by varying its direction, it will be 
 possible to make this axis a diameter of the two similar 
 ellipses. Hence generally whenever a straight line is 
 passed through a space bounded by two similar and 
 similarly placed ellipsoids, the two intercepts of the line 
 between the two surfaces will be equal. It follows that there 
 is no force in the interior of a homoeoid, whether the shell 
 is thick or thin. It may be observed that a charge 
 of electricity distributes itself over the surface of an 
 
54 NATURAL PHILOSOPHY 
 
 ellipsoidal conductor homoeoidally since there is no force 
 in the interior, or the density of the electricity at any point 
 is directly proportional to the ± on the tangent plane. 
 
 A homoeoid is obviously an equipotential surface for its 
 own field, for the potential at all interior points and at its 
 surface is constant. If we draw an equipotential surface 
 just outside and infinitely near, the distance between the 
 two surfaces will be inversely as the density a at any 
 
 point, for F "= -f- ' The density is proportional to p and the 
 
 thickness of the shell formed by the two surfaces is as — • 
 
 P 
 Hence the infinitely near equipotential surface is an ellip- 
 soid confocal with the homoeoid. If we distribute ho- 
 moeoidally on the confocal ellipsoid a mass equal to the 
 original mass, it will be an equipotential surface for its 
 own field and the flux leaving this surface will be the 
 same as the flux from the original mass. Hence if we 
 distribute homoeoidally equal masses on two infinitely 
 near confocal ellipsoids their external fields will be the 
 same. By a simple extension of the reasoning it is evident 
 that any two confocal ellipsoids on which equal masses 
 are distributed homoeoidally have identical external fields, 
 and if the masses are not equal, the force at any external 
 point is in the same direction and proportional to the mass. 
 Further, all external equipotential surfaces of a homoeoidal 
 distribution are confocal ellipsoids. This is Charles* 
 Theorem. 
 
 24. The Potential Function 
 
 Fig. 10 represents any two masses with centres of in- 
 ertia at G and G\ P and P' are any two points in the 
 masses and GG' = R, GP = r, G'P = p, GP' = p', and 
 PP' = p". The two bodies have a mutual field of energy 
 represented by their mutual potential which for any two 
 
 dyyi dyyi 
 
 elements is -, — . We wish to find the sum of the po- 
 
 P 
 
THE POTENTIAL FUNCTION 55 
 
 tentials for all the elements, thus deriving the total energy 
 of the system. The potential between the first body and 
 
 dyyt 
 an element dm' of the second body at P' is dm' S -77-, 
 
 P 
 / dm^ 
 
 and the total energy is S ( dm' S — 77- ). Taking rectangular 
 
 (,m'^p). 
 
 co-ordinates with G as origin and p' as axis of x, we have, 
 since p"2 = p'2 _ {2p' x - r^), 
 
 ^dm^^djn/2/x^y^^dm/x^^ 
 P P \ P 2 / P \ P 
 
 ) 
 
 3x2 - r2 5^3 _ 3xr2 3Sx^ - SOx^r^ + Sr* 
 
 I n fl I" o .f± "I 
 
 2p'2 ' 2p'3 ' 8p'4 
 
 Fig. 10. 
 
 Since G is the centre of inertia all the odd powers vanish 
 
 in the integration, and we have 
 
 ^dm _ ^dm ( 3x2 - ^2 35:^^ - 30x2r2 -}- Srj ,\ 
 
 p" " P' V 2p'2 "^ V^ - "^l 
 
 S <imr« is evidently a constant. S dmr2 = ;r • 
 
 S dmx2 = — ^^^r Ip', where Ip' is the moment of 
 
 inertia about p' as an axis. 
 
 The second term is therefore x-7^ Let us 
 
 suppose that the bodies are spheroids, either homogeneous 
 
 or made up of homogeneous confocal shells. Integrating 
 
 we have 
 
 _ . . 15 VA±B_±C J. ,12 
 
56 NATURAL PHILOSOPHY 
 
 In like manner 
 
 S dmx2r2 = S dmx^ + ^ 1" ^ + ^ + C _ ^n ^^, _^ 
 
 2M 
 
 35 
 
 where a and 6 are the semi-axes of the shells. The third 
 term therefore contains Ip i2, Ipi, Ip'^, with determinable 
 coefficients in the numerator and 8p'* in the denominator. 
 Every term of the order 2« will have 7pi**, /pl**"^ . . . 
 Ip'^, in the numerator and p'O-n + i) {^ -th^ denominator. 
 Hence we can write 
 
 P P \^ / 
 
 where F( ) is an algebraical series having the moment of 
 inertia, Ip, about an axis p in ascending powers in the 
 numerators, and p in ascending powers in the denominators. 
 The total potential is 
 
 •'-^('-'^^)-"-'[7 + <y)]^ 
 
 where I' is the moment of inertia of the second body 
 about R as an axis, 
 
 
 MF{^^^^dm'F{lfy 
 
 Here 7 and Ip' are the moments of inertia of the first body 
 about the axes R and p', and the primes are the moments 
 of inertia of the second body. 
 
 S dm' F (— r) is an integral equivalent to the total 
 
 (?) 
 
 mass M' concentrated at some point in the second body at 
 a distance p' from G, into the average of the function 
 
 m ■' ' 
 
 is obvious that p' must coincide with i?, for 
 
THE POTENTIAL FUNCTION 57 
 
 otherwise the forces giving the translational motions to the 
 two centres of inertia would not be equal and opposite 
 (in the same line), and by their mutuality this must be 
 the case. Hence we can write 
 
 M^ u zr/^\ kl + k , k l2-\-k I -{-k , , 
 (1), where F^-^j = -^3— + ^^ + etc., 
 
 k representing certain determinable constants. As has been 
 already shown, the forces acting upon the first body are 
 equivalent to a single force acting upon the mass M sup- 
 posed concentrated into its centre of inertia, together with 
 a couple acting about an axis through the centre of inertia. 
 Likewise the forces acting upon the second body are equiv- 
 alent to a single force acting upon the mass M' concentrated 
 into its centre of inertia, together with a couple about an axis 
 through this centre. From their mutuality the two transla- 
 tional forces are equal and opposite and the two couples 
 are equal and opposite, or their axes are in opposite 
 directions, since a couple is represented by a vector. 
 The force in any direction is the rate at which the potential 
 energy is used up in that direction, or the derivative of the 
 potential with respect to the direction. The force urging 
 
 the centres of inertia together is -j^ • If we simply turn the 
 
 ciK 
 
 bodies about R, the potential is not altered. 
 
 Considering the second body simply as a material point 
 
 of mass M', Equa. (1) becomes 
 
 V = ^ + M'f(|). (2) 
 
 R 
 
 It will be noted that the attraction in the plane of the 
 equator of a spheroid is greater than if the mass were 
 concentrated into its centre, while the attraction in its 
 axis is less, or for equal distances the attraction is a 
 maximum in the plane of the equator and a minimum in 
 the axis . It will also be noted that the motion of the centre 
 
58 
 
 NATURAL PHILOSOPHY 
 
 of inertia is the same whether we apply the forces parallel 
 to their original directions to this centre, or subject the 
 mass concentrated into this centre to the field. In general 
 this is not the case. [See Art. 8.] 
 
 If 7 is the angle which R makes with the equator of the 
 spheroid, then the couple tending to bring the equatorial 
 
 plane into coincidence with K is -r- = — -ttftt -r = 
 
 ay 2R^ ay 
 
 -^ {C — A) sin 7 cos y, since I = C COS27 + A sin2 7, 
 
 to a close approximation. 
 
 25. Rotary Motion 
 
 A disc Spins (Fig. 11) on its axis PP\ which is held in a 
 ring which can turn about a horizontal axis HH\ and the 
 
 whole turns about a vertical 
 axis VV. 
 
 If the disc is not rotating 
 and we turn it about the ver- 
 tical axis, it will simply obey 
 the turn and its axis PP' 
 will remain in the horizontal 
 plane. If it is rotating and 
 we turn the axis through a 
 horizontal angle d\l/ in the 
 time dt, the axis will not 
 remain horizontal. Let a> be 
 the angular velocity with 
 which the disc spins. This 
 velocity cannot be influenced 
 by the turn since there is no 
 couple about PP', and we 
 further suppose no friction. 
 Let C be the moment of inertia about the axis of the 
 disc, and A that about an axis in its plane. We can resolve 
 the original moment of momentum, Ceo, into the two 
 components Ceo cos dyp and Ceo sin dyp in the horizontal 
 
 Fig. 11. 
 
ROTARY MOTION 
 
 59 
 
 plane. Since no motion can cease instantly, when we turn 
 the axis of the disc through the angle <i^ in the infinitesimal 
 time, dt, the above moments will persist, and the moment 
 of momentum about the axis will be Ceo cos d\l/ and there 
 will also be a moment Ceo sin d\f/ 
 about a ± axis in the horizon- 
 tal plane. The rate of change of 
 a moment of momentum meas- 
 ures a couple about its axis. The 
 rate of change of the moment 
 about the axis of the disc is 
 Cu) cos dxp — Coi 
 
 dt 
 
 and the rate 
 
 of change about the ± axis is 
 
 ^ ,, . At the limit the first 
 
 at 
 
 becomes zero, and the second is 
 Cco^. Hence at the beginning of 
 the turn a couple is set up tend- 
 ing to bring the axis of the disc 
 into coincidence with the axis of the turn, and if the turn is 
 continued these two axes will coincide both in direction 
 and sense of motion about them. Generally, whenever a 
 rotating mass is turned about an axis ± to its rotation 
 axis, a couple is set up about an axis _L to the former two 
 axes, and this is called the gyroscopic couple. Any rotating 
 mass is a gyroscope. If t? is the angular acceleration about 
 the gyroscopic axis, Ccoxl/ = —A^ (1). This is the funda- 
 mental gyroscopic law from which all the properties of 
 rotary motion can be derived. The gyroscopic couple is an 
 internal force due to the inertia of the particles of the disc 
 and is nothing more than the moments of the centrifugal 
 forces which arise during the turn. 
 
 In Fig. 12 a lamina can turn about a horizontal axis 
 HH^ and this axis is turned about a vertical axis, VV\ 
 with angular velocity xp. A point P has co-ordinates, x, y, 
 referred to axes through the centre of the lamina, and the 
 
60 
 
 NATURAL PHILOSOPHY 
 
 radius r makes an angle ^ with the axis of %. The cen- 
 trifugal force of an element dm at P, is dmr sin (t? — ^)^2. 
 The moment of this force about the horizontal axis is 
 dmr sin (t? — ip) cos (t? — (p) xj/^. 
 
 Since 
 
 cos (p and 
 
 = sm (p, 
 
 X 
 
 r r 
 
 the total moment is 
 
 Sc^m sin t? cos t? {x^ — y'^)p — SJm (cos2t? -- s\n^d)xyp. 
 From symmetry, ll,dmxy vanishes, and we have the well 
 known result, 
 
 2<im sin t? cos t? {x'2- — y2)p = (^Q — A) sin ■& cos t?i/'2, 
 where C is the moment about the y axis and A that about 
 the X axis. 
 
 But we can get this result at once by gyroscopic prin- 
 ciples. Resolving ^ into an angular velocity yj/ sin ?? about 
 the y axis and yj/ cos t? about the x axis, we see that these are 
 two gyroscopic couples about HH\ viz., C sin ^yp. cos ^4/ 
 and A^ cos ^. ^ sin ??, opposing each other. The net result 
 is {C — A) sin t? cos t?t/'2, tending to place the x axis 
 horizontally. The gyroscopic couple is therefore merely the 
 
 - integral of all the 
 
 ^^^^^^^ f^ centrifugal moments. 
 
 j *^\P* ^^ another ex- 
 ample let us suppose 
 a sphere, Fig. 13, ro- 
 tating about an axis 
 PP' with angular ve- 
 locity CO, while PP' 
 can turn about a 
 horizontal axis HH', 
 which is held by 
 a fork which turns 
 about a fixed centre 
 at 5 in a horizontal plane, with angular velocity yp. 
 Resolve the rotation co into co cos t? about a vertical axis 
 and 03 sin t? about SO. Let SO = D. Considering a ring of 
 matter about SO, the horizontal velocity of any particle 
 
 Fig. 
 
EULER'S DYNAMICAL EQUATIONS 61 
 
 in this ring is D^j/ — rco sin t? cos <^, where <p is the angle 
 any radius makes with the vertical. The centrifugal force 
 
 is -jT- [Dxp — ro) sin ?? cos <^]2 and the moment of this force 
 
 dm r • n^ 
 
 about HH^ is -j=r Dip — roy sin d^ cos (p r cos cp. The 
 
 % 
 sum of the moments is I — Dxj/ — rco sin t> cos cp 
 
 
 
 r cos (pdcp = lirr. r'^oi sin t?i/' = mr2co sin t?i/'. The moment 
 of momentum of the ring about SO is mr'^oi sin t? and \j/ is 
 the angular velocity with which the momental axis turns. 
 Hence the integral of the centrifugal moments about 
 HH' is simply the gyroscopic couple. 
 For a disc the total centrifugal moments about HH' 
 
 are S Itt^cjj sin t?^Jr = -77—- w sin ^xp. But 7ri?2 is the mass 
 2 
 
 i?2 
 
 of the disc and -^ is ^2^ where k is the radius of gyration, 
 
 and the integral gives simply the gyroscopic couple. 
 
 For the whole sphere, using the relation r^ = R2 — k^, 
 where k is the distance of any vertical disc from the centre, 
 
 STTf* CO sm t?^ dk = — :z — — =- • o) sm ^xp. — ^— is the mass 
 
 2R2 
 
 and — ^ = k^, and we see again the identity of the gyro- 
 scopic couple with the sum of the centrifugal moments. 
 The gyroscopic principle gives at once the total centrifugal 
 moments acting about any axis. Without its use many 
 rotational problems would be practically insoluble. 
 
 26. Euler's Dynamical Equations 
 
 Let us suppose that a triaxial body is given an impulsive 
 velocity about some axis through its centre of inertia, 
 which is fixed. The resulting motion will in general be 
 unstable. For we can resolve the impulsive velocity into 
 
62 NATURAL PHILOSOPHY 
 
 coi, 0)2, 0)3 about the three principal axes and it is evident 
 that a pair of gyroscopic couples will result about each axis. 
 With due regard to signs, we can write 
 
 {B — C) 0)2003 = Acoi 
 
 {C — A) C01C03 = Bu)2 
 
 {A — B) 0)10)2 = Coii 
 
 These are Euler's dynamical equations. The mutual 
 interaction of these couples will cause the original in- 
 staneous axis to shift continually. 
 
 If, however, the original velocity were imparted about 
 a principal axis, the motion would be stable, for there 
 would be no couples. 
 
 Let us suppose a uniaxial body to be rotating about 
 some axis which is struck a sharp blow in any direction. 
 We always consider the centre of inertia as fixed. If the 
 body were not rotating it would simply turn about an 
 axis ± to the blow. But rotating, the instantaneous 
 velocity imparted by 'the blow combines with the original 
 rotation, and we have a new rotation about an axis which 
 is in the plane of the other two. The rotation axis, instead 
 of moving in the direction of the blow, in fact moves in a 
 direction ± to it. If ^ be the impulsive velocity imparted 
 about an axis ± to the rotation axis and i the angle the 
 
 new resultant axis makes with the original axis, tan i = — , 
 
 and the resultant angular velocity is V ^^ + w^- Since all 
 axes are principal axes, the new rotation will be stable. 
 
 27. Biaxial Bodies Under No Forces 
 
 If we subject a biaxial body to an impulsive couple, 
 measured by G^ about an axis making an angle t? with the 
 C axis, or axis of unequal moment, this is equivalent to an 
 impulsive moment about the C axis, together with an 
 impulsive component about a ± axis in the GC plane. 
 The instantaneous axis will lie in this plane. If o),- is the 
 instantaneous angular velocity and i the angle which the 
 instantaneous axis makes with C, while t? is the angle G 
 
TRIAXIAL BODIES UNDER NO FORCES 63 
 
 makes with C, G cos t? = Cco^ cos i, and G sin t? = Acoi sin ^. 
 Hence A tan i = C tan t? (1), and t and to,- are determined. 
 G is represented by a vector, constant in amount and 
 direction, and is called the Invariable Line. Since there 
 can be no rotation or moment about an axis _L to G, 
 the C axis and the instantaneous axis, which always lie 
 in the GO plane, cannot change their inclinations to the 
 invariable line. The motion therefore will consist of a 
 rotation of the GCi plane around the invariable line. 
 Let yp be the angular velocity of this plane. Then the C 
 axis turns about a ± axis in this plane with angular 
 velocity yp sm d^ = oji sin i, and the angular velocity about 
 the C axis is coj cos t = co. It is readily seen that the gyroscopic 
 couples about an axis J_ to the GO plane exactly balance, 
 so that there can be no motion about such an axis. For 
 the gyroscopic couples about this axis are CujxI/ sin t? and 
 Ayp sin t?. ^ cos ^, and by (1) these are equal and opposite. 
 The motion is thus completely determined. It consists of a 
 steady rotation of the CGi plane about the invariable line, 
 which is the axis of the impulsive couple, and the unequal, 
 or C, axis, and the instantaneous axis are fixed in this 
 plane. G, A, C and t? are given, and from these w, w,-, i 
 and ^ are readily found. The motion of the unequal axis 
 about the invariable line is called the Precession, and xp 
 is the precessional velocity. 
 
 28. Triaxial Body Under No Forces 
 
 When a triaxial body is subjected to an impulsive 
 couple about an axis through its centre of inertia, the case 
 becomes more complicated, li A, B, C are the principal 
 moments of inertia of such a body, in ascending order, then 
 Ax^ -\- By^ + Cz^ = 1 is the momental ellipsoid of the 
 body, and we have seen that it has the property that the 
 square of the reciprocal of any of its radii is equal to the 
 moment of inertia of the body about that radius. 
 
 Draw a radius, r, to any point on the momental ellip- 
 soid and on the plane tangent at this point drop a ± , ^, 
 
64 NATURAL PHILOSOPHY 
 
 from the centre. The equation of the momental ellipsoid is 
 
 S + S + S = ^^' + ^y + CZ2=1 (1) 
 
 Whence ^ = - -{• ^ + - = Aixi + B'^y^ -^ C^z^^ 
 p2 a* 0* c^ 
 
 r2 (^2 cos2ai + 52 cos2i3i + C^ COS27O (2), whereat /3i, 7I 
 are the direction angles of a radius referred to the principal 
 axes. If now we apply an impulsive couple, G, having p 
 as its axis, the moment of momentum of the body about 
 this particular p must remain constant throughout the 
 motion and equal to G. Resolved into its components about 
 the principal axes, G cos a = Aon, G cos jS = Boii, G cos 7 
 = Ca)3, where a, /3, 7 are the direction angles of p referred 
 to the principal axes, and coi, W2, W3 are the angular veloci- 
 ties about these axes at any instant. Hence, G^ = A^ooi^ -f- 
 52^22 + C2CO32 (3). Also, since the kinetic energy, T, must 
 remain constant, Acot^ + Bo)2'2 + CC032 = 2T — Ico,^ = 
 
 -^ (4), where I and co,- are the moment of inertia and the 
 
 f2 
 
 angular velocity about the instantaneous axis. Hence 
 coj = V 2T.r, or the instantaneous velocity is proportional 
 to the radius of the instantaneous axis. 
 
 Smce — = cos as — = cos/3i, — = cos 7^ (5) 
 
 coj CO,- a)j 
 
 . ^ Wi2 1 A2cOl2 + 52co22 + C2CO32 G^ , ,, 
 
 ^^^ '' = 2T> = 2T = 27 (^) 
 
 Hence the end of the instantaneous axis will always be in 
 the original tangent plane, which is therefore fixed and 
 called the Invariable Plane. The perpendicular, p, is 
 constant in length and direction and is called the Invari- 
 able Line. The angular velocity of the body about p is 
 
 p 27 
 coj — = -7;-, and it is therefore constant. If, then, we sup- 
 
 r Cr 
 
 pose the momental ellipsoid to roll on the invariable plane 
 preserving a constant angular velocity about p, the motion 
 of the body will be exactly represented. 
 
 The invariable line and the instantaneous axis cut out 
 
TRIAXIAL BODIES UNDER NO FORCES 65 
 
 cones in the body during the motion, and these are called 
 the invariable and instantaneous cones. Their equations 
 are found from (3) and (4), which combined are 
 
 (i4cOi2 + Bc022 + Cu)32) G2 = (A2cOl2 + B^OJz^ + €203^2) 2T . 
 
 Taking co-ordinates, x, y, z, proportional to the direction 
 cosines of either the invariable line or the instantaneous 
 axis, and eliminating coi, co2, £03 by the relations G cos a = 
 Ao)i = kGx, etc., for the invariable line, and by the relations 
 
 X = ka\ = k — , etc., for the instantaneous axis, we have as 
 
 CO,- 
 
 the equations of the invariable and instantaneous cones 
 respectively, 
 2AT - G2 ^ , 2BT - G2 ^ , 2CT - G2 ^ ^ .^^ 
 
 ~~A ^ "^ B ^ ^ C ^ ^ ^ ^^^ 
 
 A {2AT-G2) x2+B (2BT-G2) y2-\-C {2CT-G2) 02 =0 (8) 
 When 2AT, or 2CT, equals G2, these cones, which are of 
 the second degree, become two imaginary planes, which 
 however intersect in a real line. From (1) and (5) this con- 
 dition is that p = a, or p = c, or the instantaneous axis 
 coincides with the major or the minor axis of the momental 
 ellipsoid at the beginning of the motion. If 2BT = G2, 
 the cones reduce to two real planes. Here p = 6, or the ± 
 is equal to the middle axis of the ellipsoid. 
 
 The instantaneous cone intersects the surface of the 
 ellipsoid in a curve which is called the Polhode. This curve 
 is evidently the locus of all those points on the ellipsoid for 
 which p has a constant value. Its equation is found by 
 combining the equation of the ellipsoid with another ex- 
 pressing the fact that p remains constant. Hence it is 
 
 1 G2 
 
 since — = A2^2 + ^2^2 + c:222 = --. 
 
 p2 Z 1 
 
 This is the equation of a cone of the 2nd degree with its 
 apex at the centre. If p is equal to the middle axis, or — = B, 
 we have A (B — A) x2 = C {C — B) z2, which represents 
 
66 
 
 NATURAL PHILOSOPHY 
 
 two planes intersecting in the middle axis and making 
 angles with the A, B, plane whose tangents are 
 
 i 
 
 A {B - A) 
 
 Taking the equations, 
 
 C (C 
 
 1 
 
 B) 
 
 G2 
 
 = A2X2 + B2y2 + C2Z2 = ^^ 
 
 and Ax2 -\- By^ + Cz^ = 1, and eliminating y, we have as 
 the projection of the polhode on the AC plane, 
 
 A{B - A)x2 - C {C - B) 02 
 
 
 These projections are therefore hyperbolas. In like man- 
 ner we see that the projections upon the AB and BC 
 planes are all ellipses. 
 
 The case is peculiar for G^- = 2BT, or p = b. All the 
 polhodes are projected upon the AC plane as hyperbolas, 
 
 but in this case the hyperbolas 
 reduce to two straight lines in- 
 tersecting in the centre. The 
 polhodes in this case become 
 two ellipses and they are called 
 the separating polhodes, be- 
 cause all the polhodes on one 
 side of them enclose the major 
 axis, while all the polhodes on 
 the other side enclose the minor 
 axis. In Fig. 14 these polhodes 
 and various other polhodes are 
 shown. 
 
 Analyzing the motion for this 
 
 special case we get the fol- 
 
 From the fundamental equations (3) 
 
 lowing results: 
 and (4), 
 
 C0l2 
 
 0)3^ 
 
 c - 
 
 - B 
 
 c - 
 
 -A 
 
 B - 
 
 - A 
 
 Q2 - B2CC2^ 
 
 AB 
 
 Ql - B2032^ 
 
 BC 
 
 and 
 
 C02 
 
 C - A 
 B 
 
 £010)3. 
 
TRIAXIAL BODIES UNDER NO FORCES 
 
 67 
 
 Putting 
 
 V 
 
 {B - A) {C - B) 
 
 AC 
 
 k, 
 
 dc02 
 
 G2 
 B2 
 
 kdt. 
 
 TTr, ~~ W2^ 
 
 Integrating, 
 
 G + Boi2 
 
 = Ee 
 
 G — B(j02 
 
 where £ is a determinable constant. 
 Since G cos jS = ^0x2, this becomes 
 
 1 + cos /3 
 
 a 
 
 = ctn^ T. = Ee 
 
 IkGt 
 
 2kGt 
 B 
 
 (9). 
 
 1 — cos ^ 
 
 From (7) and (8), the invariable and instantaneous 
 cones become in this case two planes intersecting in the 
 middle axis and making angles with the AB plane whose 
 tangents are respectively 
 
 i 
 
 B - A 
 
 and =t= 
 
 i 
 
 B 
 
 C - B IC C - B 
 
 In Fig. 15, B, p, and I are the points respectively where 
 the middle axis, the invariable line and the instantaneous 
 axis pierce a unit _ 
 
 sphere about the 
 centre. As the in- 
 variable line de- 
 scribes its plane in 
 the body, the arc 
 Ip must always be 
 JL to the arc Bp, 
 and the angle B 
 between the invari- 
 able and instanta- 
 neous planes is constant. The right spherical triangle 
 B p I therefore always remains similar, and the problem is 
 reduced to determining the cones described in space by 
 the corners, B and I, of this triangle as it rotates about the 
 
 invariable line with constant angular velocity, •^- From (9) 
 
 it is evident that as the time increases the angle /3 ap- 
 proaches the value zero or tt. The middle axis therefore 
 
 Fig. 15. 
 
68 NATURAL PHILOSOPHY 
 
 moves so as to place itself in coincidence with the in- 
 variable line, the direction of the motion being such that 
 like poles, or like rotations, coalesce. The triangle B p I, 
 always remaining similar, finally is reduced to nothing 
 and the motion becomes a steady one about the middle 
 axis. 
 
 Differentiating (9), it is seen that the linear velocity 
 along a meridian, /3, p being the pole, is proportional to 
 sin /3, and the linear velocity ± to this, or along a parallel 
 of latitude, is likewise proportional to sin 0, since the angu- 
 lar velocity about p is constant. Hence the middle and 
 instantaneous axes, as they spiral inward towards p, cut 
 every meridian at a constant angle, and the paths are 
 what are called rhumb lines. 
 
 The polhode as it rolls on the invariable plane traces 
 out on this plane a curve called the Herpolhode. Its general 
 character can be seen from Fig. 15. It is limited by two 
 circles which it alternately touches and it is symmetrical 
 about points of tangency. For the special case where the 
 instantaneous axis is in the separating polhode, the 
 herpolhode is quite different. It is shown by the oval in 
 Fig. 15. If the directions of rotation about the middle axis 
 and p are similar, the path curves sharply towards p where 
 at an infinitely small distance it continues to approach p 
 indefinitely. If the rotations about the middle axis and p 
 are in a contrary sense, the instantaneous axis moves in the 
 herpolhode at first away from p and then, the body 
 turning over, curves around sharply from the other 
 direction towards p, where as before, from an infinitely 
 small distance, it approaches the pole indefinitely. The 
 instantaneous axis moves to the pole practically in a little 
 less, or a little more, than a quarter of a turn, but since it 
 cannot move directly to the pole and then stop abruptly, 
 it moves first to an infinitely small distance from the pole 
 and then continues its approach indefinitely. 
 
GYROSCOPES UNDER EXTERNAL FORCES 
 
 69 
 
 29. Gyroscopes under External Forces 
 
 Any rotating mass is by definition a Gyroscope. Let us 
 suppose a top, Fig. 16, rotating about its axis with angular 
 velocity co and held at an angle ??<, to the vertical. It is 
 then abandoned to gravity and we wish to find the motion. 
 The centre of inertia, G, is at a distance, h, from the point 
 of support. The moment of in- 
 ertia about its axis is C and 
 that about a ± axis through 0, 
 A. We shall take as axes of 
 reference, OG, an axis ± to this 
 in the vertical plane through 0, 
 and a horizontal axis through 
 _L to the other two. If yj/ is ^^ 
 the angular velocity of the plane 
 GOV about OF, the top turns 
 about the i/' sin d^ axis with 
 angular velocity ^ sin t> and 
 this axis turns about the OG axis with angular velocity yp 
 cos d^. We have then as the equations of motion 
 
 mgh sin d^ — Ccxjxp sin ?? + Axl/^ sin t? cos t? = A^ (1) 
 Ccoi} - ArP cos M = ADt (tp sin ??). (2) 
 
 Integrating (2), Ceo (cos ^o — cos ^) = A\p sin2 ^, (3) 
 Multiplying (1) by 4, (2) by rp sin t?, adding and integrating 
 
 Fig. 16. 
 
 Wg/t (cos t>o 
 
 ,, At?2 A^2 sin2 z? 
 
 cos^) =— + 2 • 
 
 (4) 
 
 Equa. (3) merely states that the moment of momentum 
 about OV remains constant, while Equa. (4) states that 
 the increase of the kinetic energy is equal to the work done 
 — both of which were a priori evident. These equations 
 determine the motion completely. 
 
 Let us now find the path (guided if necessary) which the 
 axis of the top must take in order that it may move from 
 some point 1 in its actual path to any other point 2 in its 
 actual path, in the shortest time possible. 
 
70 NATURAL PHILOSOPHY 
 
 ds 
 
 and 
 
 Since ^2gh (cos t?o ~ cos t?) = -7^, where ds is an element 
 
 of path described by the extremity of the radius of gyration 
 k, where mk^ = A, 
 
 dt = ^" , 
 
 ^2gh (cos ??o - cos ?>) 
 
 p ^£ 
 
 ij V2g/t (cos t?o — cos t?) 
 
 Calling the angle which this path makes with a meridian 
 
 at any point, r, ds = kdd- sec r. We shall take ?? as the 
 
 independent variable and vary s. 5ds = kdd^ sec r tan rbr. 
 
 „. s\nM\l/ ^ ^^ ^ ^ sint?^,, 
 
 Since — T7 — = tan r, 5 tan r = sec2 tBt = —jir 5a^^. 
 an av 
 
 sin z> ddyp .._. , . , . „ . 
 
 5t = —Tz z— ' Hence 8ds = k sm t? sm ra5^ 
 
 at? sec2 T ^ 
 
 1 ►T' r^ k sin t? sin T(i5iA 
 
 and 8T ^ ^ 
 
 -J 
 
 Integrating, dT = 
 
 yl2gh (cos t?o — cos t?) 
 fe sin z? sin r 5r/' |2 
 
 V2g/i (cos ^o - cos t?) |i 
 k sin t? sin r 
 
 S'H 
 
 ^2gh (cos t?o — cost?)/ 
 Since the limits are fixed the first term vanishes and the 
 condition that the path shall be the one of quickest motion 
 
 k sin t? sin r 
 
 ^2gh (cos t?o — cos t?) 
 where K is an arbitrary constant — say ^r — -. Hence 
 
 sin t? sin r = -^ J cos t?, - c ]^ (5) 
 
 VA l 2mg/j 
 
 is the equation of the curve in terms of the co-ordinates 
 t? and T. Substituting for sin r, 
 
 k sin t?(j^ k sin t?;/' 
 
 ds ~ V2g/t (cos t?o - cos t?)* 
 we have A sin2 t?^ = Ceo (cos t?o — cos t?) . This identifies 
 the curve with the actual gyroscopic path. Hence the axis 
 of the top moves naturally from one point to another of its 
 
GYROSCOPES UNDER EXTERNAL FORCES 71 
 
 path in the quickest possible time, and generally on a 
 spherical surface a body under the influence of gravity 
 moves from one point to another in the least possible 
 time if it takes a gyroscopic path. If the spherical surface 
 becomes infinitely large these gyroscopic paths become 
 plane cycloids, for sin t? becomes constant and the meri- 
 dians become parallel vertical lines, while cos z?o — cos t> 
 measures y from cos i^o- The rectangular equations of a 
 cycloid are x = a (z? — sin t?), >' = a (1 — cos t?). 
 
 Sin r = -=i^= = Vf • 
 
 or 2a sin2 r = ^^ is the equation of a plane cycloid in 
 terms of r and y. 
 
 We have already used this equation in Art. 4, where we 
 found that the cycloid is the curve of quickest passage for 
 a plane surface. We have now found that the gyroscopic 
 path is the curve of quickest passage for a spherical surface. 
 
 Taking the equation of the gyroscopic path, 
 
 . „ . WW ^ , cos t?o — cos t? 
 
 sm t7 sm r 
 
 
 ^lA 1 2mgl 
 
 since at the beginning sin r = 0, the path is at first ver- 
 tically downward. When sin r = 1 the path is horizontal 
 and this marks the maximum fall. We have for this point 
 
 cos t? = ^^''^' ^/l - (Cco)2cos\?o . (C^ 
 
 -VI 
 
 4mghA 1 2mghA \6m'^g^h'^A'^ 
 
 There is another value with + between the terms, but 
 this is inadmissible since it makes the value of cos t> 
 greater than unity. It will be noted that at this point the 
 gyroscopic couple tending to raise the top is just twice the 
 gravitational couple. After this the axis rises symmetrically 
 to its original height where it is momentarily at rest and 
 then repeats the process indefinitely. The path is like a 
 series of festoons hung upon a parallel of latitude at 
 equidistant points. If we suspend the gyroscope so that 
 it can move freely below the point of suspension we have a 
 gyroscopic pendulum. The axis executes a festoon motion, 
 
72 NATURAL PHILOSOPHY 
 
 but will never be in the nadir as long as it has the least 
 rotation. When the rotation ceases there is only a single 
 festoon hung at points 180° apart and this passes through 
 the nadir. From (4), we have in this case 
 
 t = f ^^ d^ 
 
 J ^Imgk (cos d^o — cos t>) ' 
 which, as we have seen, is the law of the ordinary pendulum. 
 The motion about the vertical axis is the precession, 
 while the motion along a meridian is the nutation. When w 
 is large, the amplitude of the vertical vibrations is very 
 small and the vibrations are very rapid, so that in high 
 spinning gyroscopes (tops) the eye cannot detect them, or 
 at most only a slight blurring. But the ear can hear these 
 vibrations and that is the cause of humming in tops. The 
 rapidity of the vibrations can be measured by the note. 
 
 With a slight nutation the festoons become very nearly 
 minute cycloids, for making these small relatively to the 
 surface is the same as making the surface very large 
 relatively to the festoons, and in either case these become 
 plane cycloids. 
 
 Otherwise, since when ^ and d^ become very small we can 
 neglect their squares and products, the equations of motion 
 become mgh sin t? — Coiyp sin t? = At? 
 
 Coi^ = ADt ()//sin ^). 
 Taking rectangular co-ordinates at the point of rest, 
 mgh sin ^ — Ccox = Ay 
 Cojy = Ax. 
 Integrating 
 
 mgh sin t? A VCcat . /Ceo A "I 
 
 * = — c^^^r- [— - ^'" [-a)\ 
 
 mgh sin 
 
 y = 
 
 
 C2w2 
 
 These are the equations of a cycloid generated by a 
 circle of radius ^^ ^ > rolling with uniform angular 
 
 velocity, —p , below a parallel of latitude. The time of a 
 
DRIFT OF RIFLED PROJECTILES 
 
 73 
 
 Fig. 17. 
 
 complete precession, or the time of a complete circuit 
 
 9 C 
 
 about the vertical axis, is ^ , so that when the rotational 
 
 mgh 
 
 velocity is high the precessional velocity is very slow. 
 
 If the peg of the top is rounded and the surface upon 
 which it rests rough, so that there is no slipping, the 
 conditions are different. If there is 
 a vertical axis (Fig. 17) through a 
 point P in the axis about which 
 the natural precession is the same 
 as the forced precession due to the 
 rolling of a small circle, c, of the 
 peg on the surface, and if the top 
 be given an impulsive velocity 
 about this axis such that the gyro- 
 scopic couple exactly balances the 
 gravitational couple about the 
 
 point P, then the motion will be stable. But in ordinary 
 motion with nutations, the peg would, with each dip, roll 
 on a larger circle, thus increasing the forced precession, and 
 the top would rise. With high rotational velocities the nat- 
 ural precession would be very slow while the forced pre- 
 cession would be rapid and consequently the top would rise 
 rapidly. Thus while a top with a peg ending in a mathe- 
 matical point cannot rise above the level from which it is 
 let go, a top with a rounded peg on a rough surface will 
 rise, and the rise is at the expense of 'the rotational energy. 
 Brennan has applied this principle of forced precession to 
 balancing a car upon a single rail {v. **The Gyroscope"). 
 
 30. Drift of Rifled Projectiles 
 
 Fig. 18 shows a rifled projectile viewed in the line of 
 flight from in front, its long axis making an angle t> with 
 the path of the centre of inertia, 0. The couple due to the 
 resistance of the air tending to restore the axis to the line 
 of flight increases rapidly up to a certain point with the 
 angle d^. The axis and the path at first coincide, but as the 
 
74 
 
 NATURAL PHILOSOPHY 
 
 trajectory deviates downward from its original direction, 
 the axis fails to follow it and a couple due to the air resist- 
 ance strives to bring it in line again.- If from any point the 
 path could continue as a straight line, the axis would per- 
 form a regular precession with its nutations about this line, 
 as indicated in Fig. 18. It would preserve a constant aver- 
 age inclination, ??, to the path, and if the restoring couple 
 
 be H sin t? and the moment 
 of inertia about the long 
 axis, C, it is seen from Art. 
 29 that the precessional 
 
 DBIFT 
 
 velocity IS — -^ and the 
 
 Co; 
 
 time for a complete preces- 
 
 2'kC(j3 
 
 sion 
 
 H 
 
 But the line 
 
 Fig. 18. 
 
 about which the long axis 
 strives to describe a regular 
 cone is constantly moving downward. When the point of 
 the projectile has reached its lowest point, P\ the line of the 
 path will be at some point, 0\ below 0, and if the path 
 should become straight from this point, the precessional 
 cone would become narrower. It is by such an action that 
 generally the long axis is kept close to the line of flight. As 
 the line of flight moves downward the angle i? is always 
 greater in the upper part than in the lower part of the pre- 
 cessional cone. This has an important frictional result. It 
 will be noted that the rotational and precessional motions 
 are such that the outer (away from 0) surface of the pro- 
 jectile in a certain sense rolls on the air and consequently 
 there is little air-friction on this surface. But the inner 
 surface (towards 0) moves against the air with both its 
 rotational and precessional velocities. The air is thus a 
 smooth surface for the outer surface of the projectile, 
 but rough for its inner surface. Since the precessional 
 velocity is much greater in the upper half of the cone than 
 in the lower half, there is a differential effect, the friction 
 
DRIFT OF RIFLED PROJECTILES 75 
 
 on the inner surface in the upper half greatly preponderat- 
 ing. The effect is the same as if the projectile were laid on a 
 partially rough surface and partly slipped and partly 
 rolled parallel to its long axis. If when viewed from behind 
 the projectile is rotating to the left — positive rotation — 
 the precessional motion will be to the right, and the "drift," 
 or the horizontal rolling on the dense underlying cushion of 
 air will be to the left. The axis rolls to the left or the right 
 according to the rifling but always remains parallel to its 
 original vertical plane. 
 
 If the downward velocity of the path is equal to, or not 
 greater than, the average downward velocity of the axis in 
 its precession, they will keep together, nearly meeting at 
 the point P'. But if the downward velocity of the path is 
 greater than this limit, so that when P reaches P' the path is 
 below this point, then we have the beginning of a "tumble." 
 
 The precessional velocity is proportional to H sin t> 
 and inversely proportional to co. There is consequently a 
 certain limit, readily calculable, beyond which the axis 
 cannot keep up with a too rapid downward curve of the 
 path. In high angle (mortar) firing, at the vertex of the 
 trajectory the curve is very sharp and at a certain limit, 
 depending upon if, w and r where r is the angle the path 
 makes with the horizontal, the projectile will tumble. 
 This limit will be reached sooner the greater the value of 
 CO, and therefore when high angles are used a great amount 
 of rifling is not desirable. 
 
 There have been many misconceptions as to rotary 
 motion. One is that a gyroscope is a "gyrostat," or device 
 which preserves its plane of rotation. If, by using a mathe- 
 matical fiction, we could conceive a body spinning with an 
 infinite velocity, then no finite couple could change the 
 direction of its axis and it would be a "gyrostat." But the 
 plane of any finitely spinning gyroscope is readily changed 
 by any couple, albeit the rate of change diminishes as the 
 rotation increases. Another misconception is that the 
 rifling of projectiles is for the purpose of keeping them 
 
76 NATURAL PHILOSOPHY 
 
 "end on." The chief advantage, however, of rifling is to 
 keep the projectile in the gun long enough to have the 
 slow burning powder develop its maximum gas pressure 
 and thus launch the projectile with a velocity otherwise 
 impossible. A spear, any long fusiform body, even an un- 
 tipped arrow, naturally keeps end on. They are, of course, 
 ** stiff er," or have less tendency to slew, the greater the 
 velocity, but any slewing tends to be corrected by the air 
 couple. For high angle firing, therefore, if it were possible 
 to delay the departure of the projectile until the full 
 pressure had been developed, without rifling, a long 
 unrified projectile would be preferable to a rifled one, 
 since it would not tumble, it would not wobble about the 
 line of flight as a rifled projectile does, and it would not 
 "drift," thus dispensing with an otherwise necessary 
 correction. 
 
 From the foregoing it is evident that the axis of a rifled 
 projectile describes a path such as that shown in Fig. 19. 
 The main curve loops downward and on this are super- 
 posed roulettes (cycloids) due to the nutation — ripples 
 as it were on the principal waves. The 
 reverberation of a shell as it passes, in 
 which beats are distinctly audible, is 
 OB/fT ^^^ ^^ these peculiar vibrations of the 
 ^ axis as it executes the major loops. 
 
 31. Kepler's Laws 
 
 If a body is revolving about another, 
 supposed fixed, and we consider both 
 Fig. 19. bodies simply as material points, then 
 
 it must remain in a fixed plane passing 
 through the two bodies, and its moment of momentum 
 about the central point must remain constant. For there is 
 no couple which could change the plane or moment of 
 momentum of the revolving body. 
 
 If it describes an ellipse about the attracting point and 
 this point is a focus of the ellipse, the attraction must vary 
 
KEPLER'S LAWS 77 
 
 inversely as the square of the distance. For, let M be the 
 
 mass of the attracting body and m that of the revolving 
 
 body. Writing the equation of the ellipse, 
 
 ^ a (1 - e2) 
 
 1 +ecosip ^^^' 
 
 where a is the major semi-axis, e the eccentricity, and <p 
 
 the angle any radius makes with the minimum radius, and 
 
 designating the constant moment of momentum, m r^ (p, 
 
 by N, we have from (1) 
 
 Ne sin ^ . .. Ne 
 
 w^ = — ^^ ^ 1 and mr = —r. rr cos ^^. 
 
 a (1 - e^y a(l — e^) ^^ 
 
 The radial force is the difference between the centrifugal 
 
 and attractional forces, or nir = mr^2 — ^y^ where / is the 
 
 attractional acceleration. Hence 
 
 47r2a^w _ Mm 
 r2T2 ~ l^r 
 
 and 
 
 The attractional force is therefore inversely as the square 
 of the distance, and the ratio of the cubes of the major 
 semi-axes to the squares of the periods is the same for all 
 bodies revolving about the same central fixed body. 
 
 This is otherwise evident when we consider that the 
 average centrifugal force must equal the average attrac- 
 tional force, the average of both these forces being that at 
 the average distance. If R is the average distance and n the 
 average angular velocity, 
 
 2ir . 72 47r2 
 w =^,and-^3 =-^ (3), 
 
 where M is the mass of the attracting body. This shows by 
 (2) that the average distance of the curve from a focus is a. 
 Otherwise, as the sum of the distances of any point from 
 the two foci is always 2a, the average distance of all the 
 points from one of the foci must, by symmetry, be a. 
 The above results are Kepler's Laws. 
 
 a ( - e^)J - 
 
 -a(l 
 
 -e2) 
 
 mr^a (1 — 
 
 e') 
 
 The period. 
 
 ^ " N 
 
 2ira2 V 1 - 
 1^ 
 
 - e^.m 
 
 72 
 n3 
 
 47r2 
 
 M 
 
 
 
 (2) 
 
78 NATURAL PHILOSOPHY 
 
 Kepler's laws, however, are only approximations, since 
 there is, of course, no such thing in Nature as a mass con- 
 centrated into a point, or a fixed body. When one of the 
 bodies has a very much greater mass than the other and 
 the distance between them is very great, as is the case for 
 the sun and a planet, these laws are very close approxima- 
 tions. Even so, each body describes an ellipse which has the 
 common centre of inertia of the two bodies for a focus. 
 Considering two bodies of masses M and m to be centro- 
 baric, or homogeneously symmetrical about their centres, 
 the ratio of their accelerations is inversely as their masses. 
 If, therefore we apply to both bodies the acceleration 
 of one of the bodies, reversed in direction, one of them 
 will be brought to rest, while the motion of the other, 
 relatively to it, will be unchanged. The acceleration of one 
 body relatively to the other is the sum of their accelera- 
 tions, and the ratio of the relative acceleration of m to 
 
 its actual acceleration is — — — . If then we fix M and 
 
 increase its mass by w, the relative acceleration of m will 
 be unchanged, and its motion relatively to M will be the 
 same as if it revolved about a fixed central body having a 
 mass equal to the sum of the masses. We must therefore 
 
 write instead of (3), 7^ = 71^-; , or the ratio of the 
 
 R^ M -\- m 
 
 cubes of the major semi-axes to the squares of the periods 
 
 of the planets is proportional to the sum of the masses of 
 
 the sun and each planet, and not simply proportional to 
 
 the mass of the sun, as Kepler's third law states. 
 
 If both bodies are biaxial, we shall see later that this 
 
 will result eventually in their revolving about each other 
 
 in a plane which contains both equators and each orbit will 
 
 be a perfect circle about the common centre of inertia for 
 
 a centre. Both before and after the merging of their 
 
 equatorial planes Kepler's third law would not be strictly 
 
 accurate. 
 
ATTRACTIONAL HARMONICS 79 
 
 32. Attractional Harmonics 
 
 In Art. 24 we found that the value of the potential 
 function between two spheroids, homogeneous or made up 
 of homogeneous confocal shells, was 
 
 ..^,„.,(_'),„@. 
 
 where F ( ) is a series with ascending powers of I and R 
 respectively in the numerators and denominators. Let us 
 suppose that the bodies are revolving about their com- 
 mon centre of inertia in circular orbits under their 
 mutual gravitation. In investigating the motion of M 
 relatively to M', which we shall consider a material 
 point, we can suppose M' to be fixed and to have a mass 
 M + M'. If 7 be the declination of M\ then the couple 
 tending to change 7, or the rate at which the potential 
 
 energy is used up about the axis of 7 is -j—, and this 
 
 gravitational couple will be of the general form 
 
   dV dT ,^ dl T.dl ^ , 
 
 G = -r- =- -a-j- -\- bl -: cl2 _ 4- etc., 
 
 07 07 ay ay 
 
 where a, 6, c, etc., are determinable constants. Since 
 I = C cos2 7 + A sin2 7 the gravitational couple will have 
 the general form 
 
 G = {a ~ bl -{- cD - dP + etc.) sin 7 cos 7 (1), 
 where a,b, c, are other constants. It will be noted that the 
 couple vanishes when R passes through a principal axis and 
 therefore altogether when the body is uniaxial. 
 
 We shall call the plane containing the two equal axes 
 the equatorial plane, and it will be convenient to resolve 
 the gravitational couple into a couple about the line where 
 the equatorial and orbital planes intersect — the nodal 
 line — and a couple about an axis _L to this in the equato- 
 rial plane. If xp is the precessional velocity of the C axis 
 about the orbital pole, we may call the first axis the z? axis 
 and the second axis the \p sin i? axis. 
 
80 
 
 NATURAL PHILOSOPHY 
 
 For the purposes of our problem it is indifferent whether 
 we consider M' fixed while M revolves about it, or M fixed 
 
 and M' revolving in the 
 same orbit. Let NAB be 
 the orbit of the attracting 
 body, E its pole, O the 
 centre of M, and ON the 
 nodal line. We shall sup- 
 pose the orbit to be a 
 perfect circle. Let the at- 
 tracting body be at A and 
 moving with constant an- 
 gular velocity, a. Taking 
 our measures from N, since 
 xj/ is small compared with 
 a, we can for a short time consider the angle NO A as 
 sensibly equal to at. y = POA, and the gravitational couple 
 is in this plane. Let the angle between the planes AOP and 
 BOP be (p. Then the 4 component is G cos (p and the xp sin t^ 
 component is G sin ^. By spherical trigonometry, 
 
 cos ^ . . ctn at 
 
 (p = and sm <p = , • 
 
 ' ~ ' ' Vcos2 d^ + ctn^at 
 
 Vl — sin2 d^ sin2 at. 
 
 cos 
 
 Vcos2 ^ -\- ctn^at 
 cos 7 = sin ?? sin at and sin y 
 
 Since 
 
 sm at 
 
 f-. 
 
 sin2 ?? sin2 at 
 
 cos2 i} + ctn^at 
 the t? component of the term kl" sin y cos y in (1) be- 
 comes kl^ sin ^ cos t? sin2 a^, and the \p sin ?> component 
 becomes kl" sin t? sin at cos a/. 
 
 Since the nutation is small compared with ??, we may 
 consider t? as sensibly constant, and since 
 
 I = A + {C - A) sin2 ?? sin2 a^ 
 the total 4 component can be thrown into the general form 
 
 AS^ = —bi sin2 at + 62 sin* at — 63 sin^ at 
 
 ^ b„ sin2« at -f etc. (2). 
 
 Likewise the xp sin ?? component can be thrown into the 
 general form 
 
 \ 
 
ATTRACTIONAL HARMONICS 81 
 
 A^p sin d^ = ct sin at eos at — C2 sin^ at cos at + 
 C3 sin^ a^ cos at ... . =±= Cn sin^^*-^ ai cos at + etc. (3). 
 sin2n at can be written as the limited series, 
 sin2» at = k — ki cos 2 at -{• kz cos 4 ai — ^3 cos 6 at . . . . 
 =t kn cos 2m a^ (4). 
 
 For cos 2 a/ = 1 — 2 sin2 a^ 
 
 cos 4 a^ = 1 — 8 sin2 at -{■ S sin^ a^ 
 and so on. 
 Conversely, cos 2n at can be written as the limited series, 
 
 cos 2n at = m — mi sin2 at + m2 sin^ at — mz sin*? at 
 
 =t mn sin^** at. (5). 
 
 Hence the t? couple has the general form 
 
 i4t? = 6—61 cos 2 a^ + ^2 cos 4 ai — 63 cos 6 a^ 
 
 =t 6„ cos 2n a^ (6). 
 
 By differentiating (4) we have the series 
 sin2«-i ^^ (.Qg ^^ ^ ^j gj[j;^ 2 at — ci sin 4 af + ^3 sin 6 a^ . . . . 
 =±=Cn sin 2m a^ (7), 
 
 and we can throw the yj/ sin t? couple into the general form 
 of (7). 
 
 Combining any two terms of the series (6) and (7) having 
 the same period, it is evident that we have an elliptic har- 
 monic motion, the axes of the ellipse being determinable 
 
 in each case. The fundamental period is -, a being the 
 
 a 
 
 angular velocity of the attracting body, and this is the 
 
 period of the first ellipse due to the two couples, — hi cos 2 at 
 
 and c\ sin 2 at. The following ellipses represent the higher 
 
 harmonics of this fundamental period, viz., 
 
 TT TT TT TT 
 — ^__ PTP 
 
 2a' 2>a' 4a' 5a' 
 
 Fig. 21 represents the first four ellipses viewed from 
 outside the orbit. The long arrow shows the direction of 
 motion of the attracting body. The first ellipse is vertical, 
 the second horizontal and so on alternately, the motion in 
 the vertical ellipses being always to the left, while that in 
 the horizontal ellipses is to the right. 
 
 Taking the case of the earth and the moon, since the 
 
82 NATURAL PHILOSOPHY 
 
 distance between these bodies is great, the gravitational 
 series (1) decreases very rapidly and the first ellipse is the 
 only one which is appreciable. Let us investigate this 
 ellipse. Taking the first term of 
 
 (M + Ml) F (|\ 
 
 we readily find that At? = -X sin t? cos t? sin2 at (8) 
 and i4i// sin t? = K sin d^ sin at cos at (9), 
 
 
 
 o 
 
 V Tl 
 
 The elliptic motion due to these two couples can be repre- 
 sented by a material point, or particle, moving harmoni- 
 cally in a vertical ellipse, to the left, with constant angular 
 
 velocity 2a. The major axis of the ellipse is ^rp- sin t>, 
 
 IX- 
 
 and the minor axis is tt?^ sin t? cos t?. The vertical veloc- 
 zGco 
 
 ity in the ellipse will be t? = — tt^ sin t? sin 2 at and 
 
 the horizontal velocity, xp sin t> = tttt- sin t? cos t? cos 2 a^ 
 If now we suppose the ellipse to move bodily to the left 
 
 zx- 
 
 with a horizontal angular velocity — 7^7=^ sin t> cos z?, the 
 total horizontal velocity of the particle will be yp sin t? = 
 
ATTRACTIONAL HARMONICS 83 
 
 K K 
 
 777T- sin ^ cos I? (cos 2 at — 1) = — 7;- sin t? cos ^ sin2 at. 
 
 Substituting the extremity of the axis of the earth for the 
 particle, the horizontal angular velocity xp sin t> will gyro- 
 scopically cause a vertical angular acceleration, 
 K sin ^ cos t? sin2 at, 
 
 and the vertical velocity, t? = — ^r^ sin t? sin 2 a/, will cause 
 
 a horizontal angular acceleration — ^^ sin t? sin 2 at. 
 
 But these gyroscopic couples are exactly equal and op- 
 posite to the gravitational couples (8) and (9). We 
 have seen that gyroscopic couples are purely internal 
 forces, representing merely the moments of the centrifugal 
 forces, which are forces of inertia. The gravitational 
 couples are the applied forces and the gyroscopic couples 
 are the forces of inertia, due to the motion. These forces are 
 exactly balanced and therefore the axis of the earth moves 
 freely (without constraint) in the first ellipse. The axis 
 of the earth executes a harmonic motion in the first ellipse 
 with constant angular velocity, 2a, in a counter clockwise 
 direction, while the ellipse itself performs a constant 
 horizontal retrograde precession about the pole of the 
 
 moon's orbit with angular velocity, ^o = — 97^ cos t?o> 
 
 where 1^0 is the constant inclination of the centre of the 
 ellipse. 
 
 In Equa. (6) there is a single unpaired term, b. This is 
 
 equal to ^0 sin ??o = — — r- sin ^0 cos t?o- The motion of the 
 zGco 
 
 mean position of the axis, t?o, thus produces a gyroscopic 
 
 couple, Cwi/'o sin t?o, which exactly balances the gravita- 
 
 tional couple for this inclination, viz., — y sin t?o cos ^oy 
 
 with a resulting constant and smooth precession of this 
 mean position, viz., the centre of the harmonic ellipse. 
 It will be noted that in deriving Equa. (6) every term 
 
84 NATURAL PHILOSOPHY 
 
 gave a harmonic not only of the order of the term but also 
 of all the lower harmonics, together with a constant 
 (zero harmonic) which is a part of 6, the couple pro- 
 ducing the constant retrograde precession. The constant 
 precession is therefore represented by a series made up of 
 
 alternating plus and minus terms, and the value derived 
 
 j^ 
 
 from the first ellipse alone, viz., xf/o = — ^r^- cos t?o, is 
 
 zGco 
 
 slightly greater than the actual value. The actual motion is 
 
 the following: The centre of the first ellipse executes a 
 
 constant retrograde horizontal precession. A point in this 
 
 ellipse moves with a constant angular velocity, 2a, in a 
 
 positive direction (to the left). About this point another 
 
 point describes the second ellipse with an angular velocity 
 
 4a in a negative direction. About the last point another 
 
 point describes the third ellipse with an angular velocity 
 
 6a, and so on, while the axis of the earth moves in the last 
 
 ellipse of all. 
 
 Whenever the couple ceases, the axis comes instantly to 
 
 rest, and starting from any position it immediately falls 
 
 into motion in that part of the harmonic ellipse necessary 
 
 to bring it to rest at the node. Considering only the first 
 
 ellipse, the inclination of the axis to the pole of the orbit 
 
 is always greatest at the nodes and least at quadratures. 
 
 That is, the axis is always at the bottom of the ellipse at 
 
 the nodes. Where two biaxial bodies revolve about each 
 
 other at a distance which is not an excessive multiple of 
 
 their diameters — and there are such instances among 
 
 heavenly bodies — not only the fundamental period but a 
 
 number of the higher harmonics would be appreciable, 
 
 constituting a veritable "Music of the Spheres." 
 
 33. Simplification of Motion 
 
 In investigating the precession of a planet we saw that 
 the gravitational couple tending to bring the equatorial 
 plane into the line joining the centres of inertia of the two 
 bodies was 
 
SIMPLIFICATION OF MOTION 85 
 
 K sm 7 cos 7 = ^3 (C — A) sm 7, cos 7, 
 
 where 7 is the declination of the attracting body. But 
 the action is mutual and an equal couple strives to 
 bring the attracting body into the plane of the equator. 
 This couple causes the plane of the orbit to precess about 
 the pole of the planet just 
 as the attractional couple y/^ / a 
 
 of the satellite causes the 
 polar axis of the planet to 
 precess about the pole of 
 its orbit. Let us suppose 
 a satellite, Fig. 22, to re- p^^^ 22 
 
 volve about a spheroidal 
 planet at a constant distance from 0. If the planet were 
 a sphere, the satellite would describe a circular orbit 
 NAN'. The component of our couple in the direction 
 of the path of the satellite is K sin 7 cos 7 cos r, where 
 T is the angle the path makes with a meridian. The 
 velocity, instead of being constant, will therefore be 
 retarded from N to A and accelerated from A to N\ 
 but the velocity in the equator will always be the same, 
 viz., the velocity for the circular orbit. It is evident, there- 
 fore, that a satellite cannot describe a circular orbit about 
 a biaxial body, unless it keeps in the equatorial plane. It 
 executes a path NBC wholly within the circular orbit, 
 and making the same angle with the equatorial plane at 
 N and C as the uninfluenced path NAN\ The node C 
 occurs before the node N' and the effect of the couple 
 is to make the nodes regress, while the average inclination 
 of the path to the plane of the equator and the average 
 velocity remain constant. The inclination of the plane of 
 the path to the plane of the equator is least at the summit, 
 B, and greatest at the nodes, where it is the constant in- 
 clination of the uninfluenced path. The motion of the orbit 
 is the same as that of a solid ring, into which we may sup- 
 pose the mass of the satellite to be uniformly distributed^ 
 
86 NATURAL PHILOSOPHY 
 
 rotating with the same angular velocity. A point on this 
 ring gives the position of the satellite at any time. The 
 motion of such a ring is obviously a precession of its axis 
 about the pole of the planet, accompanied by nutations, 
 precisely as in the case of a top. 
 
 The above is on the supposition that the satellite main- 
 tains a constant distance from the planet. Actually such a 
 condition is only possible when the orbit coincides with the 
 equatorial plane. 
 
 Actually the orbit not only precesses, but gradually 
 loses its inclination until it finally coalesces with the equa- 
 torial plane, in somewhat the same way as a plate spinning 
 on its edge on a table eventually coincides with the table. 
 A biaxial body eventually brings any revolving body 
 permanently into the plane of its equator. Rigorous proofs 
 of this have been given by Laplace and Tisserand. An in- 
 formal explanation may be found in "Popular Astronomy," 
 Sept. 1915.* Having got our satellite into the equatorial 
 plane, let us see what happens next. A body launched in 
 the equatorial plane will, in general, describe an orbit 
 which is nearly an ellipse, although not exactly an ellipse, 
 provided the planet is biaxial. Taking first the case of a 
 spherical planet of mass, M, the orbit will be an ellipse 
 
 and by Art. 32 it is readily seen that — = -rrz , 
 
 where ri is the least distance of the orbit from the planet, 
 n the greatest distance, and A^ = r2^. if now we suppose 
 the planet to be slightly flattened, the attraction at all 
 points in the equatorial plane will be increased, so that 
 starting from n the corresponding maximum distance 
 r2^ will be shorter than rz. The path will be nearly an 
 ellipse corresponding to a slightly greater mass at the 
 focus. Likewise starting from r2, the corresponding 
 minimum distance n^ will be shorter than n. 
 
 1 _ 2 (M + dM) 1 1 2 (M + dM) 1 
 
 rii A^2 r2' n' " m n* 
 
 * Some problems in Gravitational Astronomy. — The Author. 
 
SIMPLIFICATION OF MOTION 87 
 
 Whence r = and — > — (1). 
 
 That is, owing to the flattening of the planet, the satel- 
 lite describes an approximate ellipse with a major axis 
 which is slightly less than that of the original ellipse and 
 the maximum distance from the focus is decreased by a 
 greater amount than the minimum distance. Calling the 
 eccentricity of the original ellipse, e, and that of the new 
 
 1 — ^1 \ — e 
 
 approximate ellipse, e^, we have from (1) :; —   — - > -. — ; — , 
 
 1+^1 \ -\- e 
 
 ox e > el, or the new approximate ellipse is less eccentric 
 than the original one. It will be seen that in the inward 
 journey the new path is within the original ellipse, while on 
 the outward journey it is without. Thus the inward half of 
 the new path is more eccentric than the old path, while the 
 outward half is less eccentric, but on the whole the new 
 path is less eccentric. It is further evident that owing to 
 the greater eccentricity of the inward half, the minimum 
 radius will be slightly ahead of the old one, while owing to 
 the lesser eccentricity of the outward half 
 the maximum radius will be behind the 
 old one. 
 
 In other words, the major axis pro- 
 gresses at minimum distance and re- 
 gresses at maximum distance, but the 
 former exceeds the latter, so that on the 
 whole the major axis progresses, or moves 
 in the direction of the motion with each 
 revolution. In Fig. 23, the full line rep- 
 resents the original ellipse and the dotted 
 line the transformed path. We see then that a satellite 
 revolving in the equatorial plane of a biaxial planet: 1. 
 Alternately increases and decreases its eccentricity, but on 
 the whole progressively decreases it, until it revolves in 
 a perfect circle. 2. The approximate major axis, by alter- 
 nate progressions and regressions, on the the whole pro- 
 gresses. 3. The semi-major axis, or mean distance, pro- 
 
88 NATURAL PHILOSOPHY 
 
 gressively decreases until the final circular orbit is attained. 
 Better, however, than any theoretical proof is the direct 
 experimental proof which meets our eyes at many points 
 of the heavens. All the nearer satellites revolve nearly 
 in their equatorial planes in almost perfect circles, and they 
 would perform these motions exactly if it were not for the 
 disturbing action of the sun. There can be no more beauti- 
 ful experiment than the following: Suppose some power 
 able to hurl masses of matter at some planet isolated in 
 space. The planet would catch them, and winding them 
 about itself would gradually bring them all into its 
 equatorial plane, moving in nearly perfect circles. A single 
 satellite would describe an exact circle. 
 
 We shall see directly that the disturbing action of the 
 sun causes the orbits of satellites to assume a compromise 
 position between the equatorial and orbital planes of the 
 planet. The plane about the axis of which an orbit per- 
 forms its precessions is called the fundamental plane of 
 the orbit. It is not necessary that the influencing body 
 should be within the orbit, for a distant body can like- 
 wise produce a precession of the orbit. The sun causes the 
 moon's orbit to precess exactly as the earth's equatorial 
 protuberance does, and it happens that the sun's influence 
 is considerable, due to his great mass, while the earth's 
 influence is slight. We may represent such a precession by 
 a vector perpendicular to its fundamental plane, having a 
 length equal to the precessional velocity, and in the case of 
 several influencing bodies we can compound the effect by 
 compounding the vectors. Thus the precession of the 
 moon's orbit due to the sun has the ecliptic for its funda- 
 mental plane, while the precession due to the earth has the 
 earth's equatorial plane for its fundamental plane. The 
 resultant precessional axis lies in a plane containing the 
 axis of the ecliptic and the earth's polar axis, and inclined 
 to the former about 12'. The inclination of the moon's 
 orbit to this resultant axis is about 84° 40'. Hence as the 
 moon's orbit rotates about this resultant axis, its in- 
 
SIMPLIFICATION OF MOTION 89 
 
 clination to the ecliptic varies from a maximum of 5° 20' to 
 a minimum of 4° 56'. 
 
 This effect of the equatorial protuberance of a planet in 
 bringing a satellite into its plane and then destroying its 
 eccentricity, is very strong when the satellite is near the 
 planet. It is strikingly shown in the case of the satellites 
 of Mars and of all the nearer satellites of our system. 
 The equatorial planes of the planets are, of course, con- 
 stantly shifting, due to planetary precession, but they 
 carry their nearer satellities with them practically the same 
 as if their orbits were rigidly attached. 
 
 These gravitational effects all exemplify a general 
 principle in Nature which we may call the Simplification of 
 Motion. There is everywhere a tendency to reduce com- 
 plicated and irregular forms of motion to simpler and more 
 regular forms. By the development of gyroscopic couples, 
 two or more rotations tend to fuse into a single rotation. 
 This tendency may result only in an oscillation about the 
 position of fusion (equilibrium) but frictional forces 
 eventually effect the fusion. The motion of a triaxial body 
 with its instantaneous axis in the separating polhode is an 
 example of the simplification of motion. Tidal forces tend 
 to equalize rotational and revolutional motions and even- 
 tually do equalize them — this being the simplest form of 
 such a double motion. We shall see directly that all ro- 
 tational planes tend to coalesce with revolutional planes. 
 
 The first, second, and third satellites of Jupiter are an 
 example of the harmonizing of motions. Considering a 
 revolution as a vibration, and circular orbits are composed 
 of two simple harmonic motions perpendicular to each 
 
 other, the frequencies of these vibrations are ^, ^, and -=-, 
 
 i 1 i 2 ^3 
 
 where T is the period. By their mutual interactions, they 
 have been able to bring their frequencies into a simple 
 harmonic relation. The frequencies are very nearly as 
 
 1, ^> J- The five inner satellites of Saturn have frequencies 
 
90 NATURAL PHILOSOPHY 
 
 nearly as ^^1, 1, 1 . . . 1 . . . ^. A vibrating 
 
 body not only tends to set up harmonic vibrations in other 
 bodies, but when that is impossible and the shape is 
 changeable, actually tends to shake them into forms capa- 
 ble of such harmonics. A rigid body can only respond to 
 certain fixed frequencies, but an elastic body may adjust 
 itself to the proper frequencies. 
 
 The mutual tendency of the orbits of our system is to 
 coalesce into a single plane, and, given time enough, they 
 will eventually coalesce into the Invariable Plane of the 
 system. And nowhere is there an opposite tendency. Simple 
 and regular motions never degenerate into complex and 
 irregular forms, for the simpler the motion the more stable 
 does it become, and all irregular and complex forms are 
 essentially unstable. 
 
 34. Effect of Moon's Orbital Precession on Earth's Axis 
 
 Owing chiefly to the sun's action, the moon's orbit per- 
 forms a complete precession, with the ecliptic as its 
 
 fundamental plane, in about 
 
 /^ ^s. 18^ years. This in turn exerts 
 
 \ an influence on the earth's 
 ^ \ axis which we shall now ex- 
 
 amine. Let A, Fig. 24, be the 
 position of the earth's axis, 
 C the pole of the ecliptic, d^ 
 the angle CA, B the pole of 
 the moon's orbit, and CB the 
 constant angle, a. Actually d^ 
 is about 23° 27' and a is nearly 
 5°. It is evident that we can 
 effect the steady retrograde 
 precession caused by a revolving body, by fixing half its 
 mass at the pole of its orbit and supposing it to exert a 
 repellent instead of an attractive action. We have then to 
 consider the action of a body having half the moon's 
 
MOON'S ORBITAL PRECESSION 91 
 
 mass and moving in a retrograde direction in the small 
 circle with constant angular velocity, —b. Let the angle 
 BA be c. Then the gravitational couple is —K sin c cos c. 
 If the angle CAB = A, the & and rp sin t? couples are 
 
 — K sine cos c cos A and K sin c cos c sin A . 
 
 Since yp, the precession of the earth's axis about C, is 
 
 small compared with 6, the angle ACB = C, measured from 
 
 a position of conjunction, will for a short time be sensibly 
 
 - sin A sin a ... - 
 equal to — bt. — — 7^ = ~. — (1) and cos c = cos a cos t} + 
 ^ sm C sm c 
 
 sin a sin ^ cos C (2). 
 
 Hence the 
 
 t^ couple is — K cos c V sin2 c — sin2 a sin2 C. 
 
 sin2 c — sin2 a sin2 C = (cos a sin ^ — sin a cos t? cos C)2^ 
 
 and the t? couple is 
 
 — K" cos c (cos a sin t? — sin a cos i} cos C) . 
 
 Substituting the value of cos c from (2) this becomes 
 
 A^ = —K cos2 a sin t? cos t? + i^T sin a cos a cos 2 t? cos C + 
 
 K sin2 a sin ?? cos t? cos2 C (3). Likewise 
 
 Axp sin i> = K" sin c cos c sin A = K sin a cos a cos z? sin C + 
 
 K' sin2 a sin z? sin C cos C (4) . 
 
 Our equations of motion therefore are 
 
 K sin2 a sin z? cos t> cos2 bt -]- K sin a cos a cos 2 t? cos 6i — 
 
 K cos2 a sin t? cos t? - Ccoxp sin t? + i4i/'2 sin ?? cos t> = At?. (5) . 
 
 — K" sin2 a sin t? sin 6^ cos bt — K sin a cos a cos t? sin 6^ + 
 
 Cco^ - Ai^ cos M = A)A sin t? (6). 
 
 [We have dropped the angles A and C and these symbols 
 now resume their usual significance.] 
 
 4 and \p are so small that we can neglect second powers and 
 T? is sensibly constant. The motion is so small that we can 
 use X in place of xp sin t? and y in place of ??. In other words 
 we can use rectangular in place of spherical co-ordinates. 
 Hence our equations of motion become, 
 K sin2 a sin t> cos t? cos2 bt -{- K sin a cos a cos2 t? cos 6^ — 
 
 K cos2 a sin t? cos t? — Co3X = Ay (7) 
 and — K sin2 a sin t? sin 6/ cos bt — K sin a cos a cos i> sin 6/ + 
 Cw>' = Ax (8). 
 
92 NATURAL PHILOSOPHY 
 
 Integrating (8), Ccoy — A {x — Xo) = 
 
 r^ • o • « sin2 bt 
 K sm^ a sm t? 
 
 K sin a cos a cos ?> 
 
 26 
 ( cos bt - 1) 
 
 6 
 
 Since x is small compared with co, we can write this 
 
 ^ r^ • o • o sin2 bt , 
 C(i}y = A. sin2 a sm t? — yr h 
 
 T^ ' (1 ~ cos bt) .^. 
 
 A sm a cos a cos t> ^ r (9). 
 
 Integrating (7), 
 
 ^0 
 
 ^ rx • -> • o Q I ' . (sin6/cos60\ , 
 Ccoic = K sm2 a sm I? cos t? (7^ H ^T ^ ) + 
 
 K sin a cos a cos 2 t> — r — — K cos2 a sin ^ cos ??. / (10). 
 
 Plotting the curve from these equations we find that it 
 has the shape given in Fig. 24. 
 
 Starting with the origin at the time when the poles 
 are on the same celestial meridian (conjunction), the 
 inclination of the earth's axis to the pole of the ecliptic is 
 here a minimum. After this it increases until a maximum is 
 reached with bt = x. It then regains its former minimum 
 when bt = It and the two poles are again in conjunction. 
 The path of the earth's axis is thus an unsymmetrical 
 wavy curve. There are about 1400 such complete waves in 
 every complete precessional circle of the earth's axis about 
 the pole of the ecliptic. The variation of t?, or the depth of 
 the curve is about 9''. 
 
 35. Effect of the Moon*s Orbital Precession on her 
 
 Own Axis 
 
 This action upon the earth's axis, due to the shifting of 
 the moon's orbit, is purely reciprocal. 
 
 The earth's attraction upon the moon's equatorial pro- 
 tuberance causes the moon's axis to precess (retrograde) 
 about the pole of her own orbit. As far as the precessional 
 effect is concerned, it is a matter of indifference whether 
 
EFFECT ON HER OWN AXIS 93 
 
 the moon revolves about the earth or the earth revolves 
 about the moon in the moon's orbit. We can cause the same 
 precessional effect upon the moon's axis by supposing 
 half the earth's mass to be at the pole of the moon's orbit 
 exerting a repulsional instead of an attractional action. 
 If then in Fig. 24 we suppose A to be the pole of the moon's 
 orbit moving with a constant retrograde precession, — 6, 
 and half the earth's mass to be at this pole, while the 
 moon's axis is at B, the problem, though reversed, is 
 exactly similar to the previous one. The action of the 
 earth will be to cause the moon's axis to precess in the 
 small circle, B, with nutations, forming a wavy curve 
 precisely as in the other problem. Calling now the angle 
 CA, ??, and the angle CA, a, the equations of motion are 
 
 K sin2 a sin t? cos t? cos2 {y^/ — hi) -f- 
 K sin a cos a cos 2 ^ cos (i/' — bt) — K cos2 a sin t^ cos x? — 
 Ccot/'sint? = A^ (11) and 
 K sin2 a sin ?? sin (^ — bt) cos (yj/ — bt) -f 
 K sin a cos a cos ^ sin (^ — bt) + Ccor? = Axf' sin r> (12), 
 where K and rp now refer to the moon. The angle, xf/ — bt^ 
 is the difference between the precession of the moon's axis 
 and the precession of the pole of her orbit. 
 
 Calling ^ — bt,a, the equations of motion can be written 
 D cos2 a + Ecosa - F - Cco^ sin t? = At? (13). 
 G sin a cos a + ^ sin a + Ccoz? = A4^ sin t? (14), 
 where D, E, F, etc., are determined constants. The natural 
 independent precessions of the two poles we are consider- 
 ing, viz., xj/ the precession of the moon's axis and bt the 
 precession of the pole of her orbit, are not the same. 
 However, they are forced into coincidence by a peculiar 
 action which we shall now discover. The natural pre- 
 cessions being different, one pole will eventually overtake 
 the other and at some time a will be momentarily zero, 
 and at some other time momentarily tt — conjunction or 
 opposition. From Fig. 24, we see that in these positions 
 !? is momentarily zero. 
 
 Let us suppose that a has become tt, the two poles 
 
94 NATURAL PHILOSOPHY 
 
 being at their maximum distance apart, with C the pole of 
 the ecliptic between them. From (14), the precessional 
 acceleration, \j/ sin t?, is here zero, and the precessional 
 velocity is momentarily constant. But ^ and h not being 
 equal, the moon's axis will directly either get ahead of or 
 lag behind the orbital pole. If the angle a becomes negative, 
 from (14) the precessional acceleration becomes negative 
 and a couple is brought into play tending to bring the 
 moon's axis back into coincidence (opposition) with the 
 orbital pole. If the moon's axis gets ahead and the angle a 
 becomes positive, from (14) a positive couple arises tending 
 to turn the moon's axis back into coincidence with the 
 other pole. 
 
 There is a limit beyond which this regulatory couple 
 could not overcome the difference between the natural 
 velocities, but in the case of the moon, her mass being 
 slight, the couple is well within this limit. 
 
 When a is zero, if the moon's axis gains from this 
 position, a becomes negative and a negative couple arises 
 which tends to increase the gain still further, while if the 
 moon's axis lags, a becomes positive and a positive couple 
 arises which tends to set it still further back. Hence, 
 when the two poles are in conjunction they exert a mutually 
 repellant action and are in unstable equilibrium, while 
 when they are in opposition they are in stable equilibrium. 
 
 We have here another case of the simplification of mo- 
 tion. Instead of pursuing an irregular motion with two 
 independent precessions, the poles fall into step 180° 
 apart, and the motion is afterwards performed as if the 
 moon and her orbit were rigidly connected, moving to- 
 gether as a whole in the symmetrical position where the 
 moon's axis, the axis of her orbit and the axis of the 
 ecliptic all lie in one plane. The moon's axis thus moves 
 with a nearly constant precession and with a nearly con- 
 stant inclination to the pole of the ecliptic — practically 
 a Poinsot motion, or a motion under the action of no 
 forces. 
 
EFFECT ON HER OWN AXIS 95 
 
 The same regulatory couple exists for the earth's axis in 
 the problem previously considered, and there is a tendency 
 to force the earth's precession to keep step with the pre- 
 cession of the moon's orbital pole, but the masses being 
 reversed, the regulatory couple is unable to force the 
 earth's axis to the proper velocity and it constantly lags 
 behind. The distortion of the curve in Fig. 24 plainly 
 indicates the effort which the orbital pole makes to carry 
 the earth's axis along with itself. 
 
 This peculiar motion of the moon's axis has been 
 exactly confirmed many times by observation. Cassini 
 first discovered it in 1675 by observation, and it is known 
 as Cassini's Theorem. It is usually stated thus: "The 
 plane of the moon's orbit, her equatorial plane, and a 
 plane through her centre parallel to the ecliptic, always 
 intersect in the same line, and the ecliptic plane always 
 lies between the other two." 
 
 In all revolving systems, both the central and the satel- 
 litic bodies always tend to fall into a Cassini motion, and 
 the smaller bodies generally acquire such a condition at an 
 early stage. It is quite certain that all the nearer satellites 
 of our system execute Cassini motions, and from tidal 
 forces all the nearer, and probably also the remoter ones, 
 perform their rotations and revolutions in the same 
 period. There is however no connection between the two 
 phenomena except in so far as a low rotational velocity 
 favors the action of the regulatory couple. 
 
 A certain historical interest attaches to Cassini's 
 theorem. Shortly after this peculiar motion was discovered, 
 it was perceived that there must be some cause and 
 an explanation was eagerly sought. In 1754, D'Alembert 
 attempted a solution without success. Finally in 1764 the 
 French Academy offered a prize for the discovery of the 
 cause and this prize was won by Lagrange in 1780. His 
 solution, however, was only a partial one. Lagrange proved 
 that if the moon is triaxial with the axis of least moment, 
 always pointing nearly towards the earth, then, such a 
 
96 NATURAL PHILOSOPHY 
 
 condition of the three planes once existing, it would 
 persist. Routh, in his "Advanced Dynamics," has given a 
 proof along similar lines. Both of these proofs postulate 
 that the axis of least moment shall always point approxi- 
 mately towards the earth. But we have seen that the 
 coincidence of the rotational and revolutional periods is 
 not essential and the body need not be triaxial. In fact we 
 have supposed the moon to be biaxial. 
 
 36. Glacial Epochs 
 
 We have seen that two bodies revolving about their 
 common centre of inertia are subject to tidal forces 
 tending to tear each body apart from its centre towards 
 and away from the other body. They are thus lengthened 
 in the direction of the line between them and compressed in 
 a direction perpendicular to their orbits. If the rotation 
 and revolution are not the same, the matter of the body in 
 rotating through these body tides is subjected to a kneading 
 process which reduces its rotational velocity by transform- 
 ing rotational energy into heat. In the case of the earth 
 these body tides are not inappreciable. They are slight 
 but they certainly exist, and the continuous operation of 
 even a slight action for immense periods of time has, as 
 we shall see, far-reaching effects. 
 
 The earth is rotating at all times about two axes — the 
 polar axis and the precessional axis which is perpendicular 
 to the former. As these rotations have to be executed 
 through the tidal distortion, they are constantly being 
 opposed. The angular velocity of the tide is the orbital 
 angular velocity, a, of the attracting body — sun or 
 moon — and for both bodies the average axis of the tide is 
 perpendicular to the ecliptic. If H is the tidal effect, or 
 couple, we may suppose it decomposed into two tides, 
 H cos d^ about the diurnal axis and H sin i? about the pre- 
 cessional axis. The effect of either tide is proportional to 
 the difference of the tidal and rotational velocities, or 
 
GLACIAL EPOCHS 97 
 
 the couple about the diurnal axis is proportional to H cos t? 
 (a — co), while that about the precessional axis is pro- 
 portional to H sin ?? (a — ^ sin d). The former couple is 
 negative and tends to reduce w to a, while the latter is 
 positive and tends to reduce the negative precession, 
 yj/ sin t?. 
 
 The constant negative precessional velocity, yj/ sin t?, 
 which is just sufficient to balance the average gravitational 
 couple and maintain the inclination constant, we have 
 
 found to be — j^ sin t? cos t?. If we reduce this negative 
 
 velocity by braking, or accelerate positively, it will no 
 longer be able to support the gravitational couple and the 
 axis will yield in part to this couple. In the case of a top, 
 which is precisely similar, if we reduce the precession by 
 the slightest amount, it begins to fall, and if we abolish 
 the precession altogether it falls exactly as if there were no 
 rotation. The effect of the tidal brake is that the earth 
 never has quite the full amount of precession, or the free 
 precession, necessary to maintain its inclination constant, 
 and the axis slowly falls away. 
 
 We may therefore divide the motion into two parts, 
 viz., the actual precession combined with nearly all, but 
 not quite all, of the gravitational couple, resulting in a 
 precessional motion with constant inclination, together 
 with an extremely minute gravitational couple which is 
 unbalanced by any precession and which results in a 
 pendulation through the pole of the attracting orbit, 
 exactly as if there were no rotation. 
 
 Regarding it from another point of view, we may 
 consider the actual precession to be a free precession to- 
 gether with a minute precession in the opposite, or posi- 
 tive, direction, the algebraic sum of the two being the 
 actual precession. The motion can thus be divided into 
 two parts — the free precession which exactly balances the 
 gravitational couple and which would maintain the in- 
 clination with no other forces, together with a minute 
 
98 NATURAL PHILOSOPHY 
 
 direct, or positive, precession unbalanced by any gravita- 
 tional couple, which causes the axis to pendulate through 
 the pole of the ecliptic, exactly as in the case of the gyro- 
 scopic compass. Whether regarded as an ordinary gravita- 
 tional pendulum, or as a gyroscopic pendulum, the motions 
 are equivalent. 
 
 Considering the earth as absolutely rigid, its axis 
 would precess in a small circle about the pole of the 
 ecliptic at a constant average inclination forever. But the 
 sweep of the tide is equivalent to a minute positive couple 
 about an axis ± to the ecliptic, with the result that the 
 precessional curve is not exactly re-entrant but gradually 
 spirals in towards the pole. 
 
 Let us consider the following problem which is similar 
 to, but not identical with, the actual case. We shall con- 
 sider the earth to be absolutely rigid (non-def ormable) , 
 and a constant positive couple, H, is applied about an 
 axis through its centre perpendicular to its orbit. Let co 
 be the rotational velocity of the earth at the beginning of 
 an epoch and a>3 that at any subsequent time. The average 
 gravitational couple for a complete revolution we have 
 
 found to be — -r- sin ^ cos t?, or half the maximum couple. 
 
 We can effect the same average precessional motion by 
 placing the half mass of the attracting body at the pole of 
 its orbit at a distance equal to its average distance, and 
 supposing it to repel instead of attracting. 
 The equations of motion are, 
 
 - y sin t? cos t? - Cco3 i/' sin t? + Ayp^ sin t? cos t? = A^ (1) 
 
 if sin T> + Coi3 ^ - A>P cos m = ADt (rp sin t>) (2) 
 Hcost? = CDtc^s (3) 
 
 From (2) and (3) we derive the momental equation, 
 
 Ht = Cco3 cos i^ - Ceo cos t?o + A\l/ sin2 t? (4), 
 which states that the increase of the moment of momentum 
 about the axis of the couple is measured by the time 
 integral of the couple. From (4), 
 
/ 
 
 GLACIAL EPOCHS 99 
 
 ZJf2 C2 
 
 ^ Sin2 Mt = -^ Jrf (^3^ - 0)2) + Ceo COS t?o^, 
 
 C(x)3 COS t?(i^ = ^^ (CO32 — co2) . 
 
 From (1) and (2) we have the energy equation, 
 
 ■J (sin2 t?o - sm2 1?) + -^j- - ^ (C032 - w2) H j—± H = 
 
 y (t?2 + (^Sin^)2). (5). 
 
 It is evident that the axis spirals in towards the pole with 
 a negative precession and at the pole, Ht = C{ca3 — w cos t?o), 
 while the value of t?2 at the pole is 
 
 t?2 = ^ sin2 t?o H- ^ co2 sin2 t?o. 
 
 Since K' is small compared with w, we may write 
 
 C 
 t> = -J CO sin t?o. 
 
 Thus the polar value of t> depends only upon co and is 
 independent of any intermediate values, C03, of the rota- 
 tional velocity. If there were a couple retarding the rota- 
 tional velocity, as in the actual case, instead of an ac- 
 celerating couple, the polar value of t? would be the same. 
 We shall now start from the pole, as a new epoch, with a 
 velocity, 
 
 C 
 
 & = -T- (i) sin i?o- 
 
 Let o)p be the rotational velocity at the beginning of this 
 epoch. The momental equation is now 
 
 Ht = Cco3 cos d^ - Co)p + A\l/ sin2 ?? (6). 
 
 It is evident that as the axis spirals outwards the preces- 
 sion will be direct, or in a positive direction. Equa. (5) 
 now becomes 
 
 y^^2 -^sin2 t?o) +y (v^sint?)2. (7). 
 
100 NATURAL PHILOSOPHY 
 
 Putting the value of Ht from (6) in (7) and making t? = 0, 
 
 Ayj/ sin t? cos t> = Cco3 sin t? =±= OcoZ sin2 t?o j- sin2|? V (8) . 
 
 Since K is small compared with oj, we can write 
 
 i4^ sin t? cos t? = Caj3 sin t? =*= Ceo sin t?o. (9). 
 This is the condition for the extreme outward swing. It is 
 evident that the precession will still be positive when 4 be- 
 comes zero. 4/ sin t? cos t? and C03 sin t? are the components in 
 the orbital plane of the rotational velocities about their 
 respective axes. These components are about the same 
 axis but in opposite directions and therefore have different 
 signs. If we consider 1/' sin d^ cos ?? negative, then 0)3 sin t? is 
 positive and for the right member of (9) to be negative we 
 must use the lower sign and Cwz sin t? < Ceo sin t?o- If we 
 consider ^j/ sin t? cos d^ positive, then Cco3 sin t? is negative 
 and for the right member to be positive, we must use the 
 upper sign and CC03 sin d^ < Cm sin t?o- But C03 > co, whence 
 sin ^ < sin t?o- Consequently the axis starts on its second 
 swing towards the pole from a nearer position than on the 
 first swing. It will be noted that in Equa. (9) the rotational 
 velocity 03 p for the beginning of the epoch does not appear. 
 Consequently if during the outward swing the rotational 
 velocity were retarded, as is actually the case, instead of 
 being accelerated, the motion would be similar. Any 
 variation of the velocity about the polar axis does not 
 influence the direction or general nature of the motion 
 about the other axes: it merely modifies slightly the 
 amount of such motions. 
 Making^ sin t? = 0, we have 
 C 
 
 '- A 
 
 (co2 sin2 t?o - W32 sin2 ^) V. 
 
 The point where the precession becomes retrograde is 
 therefore within the original starting point and it is evident 
 that the axis will finally come to rest _L to the orbit. 
 
 The motion is represented diagrammatically in Fig. 25. 
 The full curve represents the retrograde spiral inward 
 and the dotted curve the direct spiral outward. At A the 
 
GLACIAL EPOCHS 101 
 
 axis has ceased going outward and at B the precession 
 becomes retrograde. Each inward spiral is begun succes- 
 sively nearer to the pole. If the couple, H, is very small, 
 the period of each swing is very great. 
 
 The actual case of the earth, while generally similar, 
 differs in some respects from the preceding problem. The 
 diurnal rotation, instead of being 
 accelerated, is retarded, but the ,,.J^ 
 
 precessional rotation is acceler- ■>^<jmr7*\*»^ 
 
 ated, as in the problem. We do /^><!^^^^rSv\\'^ 
 not know the exact expressions for //'///'Ciir\*\\\\\ 
 the couples about the two axes, I M • f:''<^ U H i I I 
 and if we did the equations of \\ \^^ V^O-^'V// / 
 motion would probably not be in- V\ vV>--*^^^v'* 
 tegrable. However, the general >v»o--IIII^--^/ 
 
 nature of the motion is evident ^^^ 1 
 
 and it must be like that in the pre- ^^ ^^ Q 
 
 ceding problem. In Equa. (1) the A 
 
 factors governing the polar motion pj^^ 25 
 
 of the axis are the gravitational 
 
 couple and the gyroscopic couple, — Ccoa^ sin t?. When the 
 precessional and diurnal motions are both accelerated it is 
 evident that the axis gets nearer to the pole with every 
 swing. However, whether the amplitudes of the swings 
 successively decrease or increase depends upon the relative 
 variation of the two factors, ws and i/' sin t?, — the rotations 
 about the two axes. If 003 decreases in a relatively greater 
 proportion than the precession is accelerated, the am- 
 plitude of each swing will become progressively greater 
 until the positive end of the axis of a planet may be brought 
 to the other side of the ecliptic, and its diurnal rotation 
 will appear to be retrograde. 
 
 We may consider the nearer satellites of a planet as 
 rigidly attached, dynamically, to its equatorial plane, so 
 that the orbits of the nearer satellites will be "tipped over" 
 with the planet, and their revolutions will appear to be 
 retrograde. The fact that such motions occur in our solar 
 
102 NATURAL PHILOSOPHY 
 
 system is therefore not an argument against the nebular 
 hypothesis. The less the density and rigidity of a planet, 
 the more likely is it that its diurnal rotation will be slowed 
 down disproportionately to the tidal acceleration of its 
 precession, resulting in an extreme pendulation. If, in the 
 case of the gyroscopic compass, the precession remains 
 practically constant while the rotational velocity steadily 
 decreases, an extreme pendulation will result, even through 
 the opposite pole. 
 
 The problem as treated here does not take account of the 
 fact that besides the breaking action of the tide, the earth 
 is distorted^ so that the tidal mass is actually performing an 
 independent rotation in the plane of the ecliptic. This 
 gives rise to a moment of momentum about the axis of the 
 ecliptic equal to the mass of the tide, into the square of its 
 distance from the centre, into the orbital velocity. This is 
 the same as if the earth were perfectly rigid and had an 
 added moment about the orbital axis converting it into 
 a gyroscopic compass. The effect is very slight, but 
 additive to the breaking effect. 
 
 If the earth were a perfectly rigid body, it would main- 
 tain its inclination constant, but perfectly rigid bodies do 
 not exist. The motion of an elastic body is different and, 
 given sufficient time, the results are widely different. 
 There is not the least doubt that the earth executes pendu- 
 lations through the pole of the ecliptic, and there is abun- 
 dant evidence that it has executed several such swings in 
 the past. The extent of glaciation about the poles at any 
 time is simply a function of the axial inclination. With the 
 axis J_ to the ecliptic there would be no ice anywhere and a 
 genial climate would exist even at the poles. There is 
 undoubted evidence that a subtropical flora flourished near 
 the poles in a comparatively recent past. This is positive 
 proof that the axis at that time had little inclination, 
 for this flora could not have flourished without continuous 
 light as well as heat. With the present inclination there are 
 extensive ice caps at the poles and with a few degrees more 
 
GLACIAL EPOCHS 103 
 
 of inclination these caps would extend to twice their 
 present area, as happened during the last extreme glacia- 
 tion. 
 
 The period of the swing must be enormously great. 
 What it is we do not know and perhaps centuries must 
 elapse before we have any definite knowledge. 
 
 It is stated that Eratosthenes (B.C. 250) found the 
 inclination to be 23° 51' 20" and that Hipparchus (B.C. 
 120) found it 23° 51'. The determination of this angle 
 requires a high state of contemporaneous civilization. 
 Only a few centuries ago such determinations were no- 
 where possible. Shortly before and after the Christian era 
 there were astronomers who could attempt it. Going far- 
 ther back we come to another stage of barbarism and 
 beyond this in a very remote antiquity, we come to the 
 Pyramid builders, who were astronomers of a high order. 
 Their time has been estimated all the way from 3000 to 
 13,000 B.C. and even more, but the simple fact is we do 
 not know when they lived. 
 
 These builders have left us a peculiar angle in their 
 oldest pyramids. This angle is 26° and its double 52°. 
 The slopes of all the faces are 52° and the inclination of all 
 the passages, whether descending or ascending, is 26°. 
 The selection of this particular angle and its constant re- 
 petition could not have been accidental. They undoubtedly 
 had measured the inclination of the earth's axis to within a 
 minute. This is the one great angle in all nature which 
 would impress itself upon intelligent men. There is no 
 other predominant angle for an earth-dweller. There is a 
 presumption then that this angle was the inclination of the 
 earth's axis at that time and that its double, or 52°, was 
 the breadth of the then tropical zone. But, of course, this 
 is only a surmise.* 
 
 *r. Popular Astronomy. Dec. 1916. 
 
104 NATURAL PHILOSOPHY 
 
 37. The Earth's Surface 
 
 The rigid demonstration of a mathematical proposition 
 is something which must be eternally true. There are, 
 however, many cases where we cannot hope to arrive at 
 exact results, but only at probable truths, or even at 
 possible truths. It is thus legitimate to speculate, pro- 
 vided we always keep carefully in mind the distinction 
 between exact knowledge and surmise. In the present 
 article we shall allow ourselves to speculate upon what 
 part known forces may have had in shaping the earth's 
 surface. 
 
 The earth is progressively denser from the surface to the 
 centre, and layers of equal density are ellipsoids. The 
 ellipticity, or deviation from sphericity, of these surfaces 
 increases from the centre, where it is zero, to the surface. 
 The principal moments of inertia of these shells vary 
 therefore, and the quantity, C — A, upon which the 
 precession depends, increases to the surface. The pre- 
 cessional rates of the different shells being different and 
 the interior being plastic, the precession is not executed as 
 a whole, as it would be in a perfectly rigid body, but there 
 is a tendency for some of the outer shells to move over 
 each other. Since it is not possible for a spheroidal shell 
 to turn about its long axis over an enclosed shell very far, 
 such motion must be very limited. There result, however, 
 readjustments which may be smooth and regular, or 
 occasionally effected suddenly. Thus earthquakes arise 
 which, though actually very slight movements, seem to 
 observers like men to be of extraordinary intensity. It is 
 evident that if the earth were isolated such phenomena 
 could not arise and their occurrence points plainly to 
 the action of external bodies. 
 
 The effect of the shifting of large masses over the earth's 
 surface is an interesting problem. The aqueous vapor in the 
 atmosphere is a small but appreciable fraction of the 
 earth's mass. Ordinarily the weight of the atmosphere is 
 
THE EARTH'S SURFACE 105 
 
 distributed in a fairiy even manner over the earth's 
 surface, but if during the beginning of a glacial epoch the 
 aqueous content going poleward is locked up there and not 
 allowed to return, a disturbance of equilibrium results. 
 The first result of a concentration of matter towards the 
 poles is a diminution of the moment of inertia about the 
 earth's axis with a corresponding increase in the rotational 
 velocity, since the moment of momentum must remain 
 constant. A secondary result, which is corrective of the 
 former, is a bulging of the equatorial regions, due to the 
 increased rotation and the increased weight at the poles, as 
 the earth strives to regain its former potential surface. 
 These effects are extremely slight, but an inequality may 
 remain uncorrected for some time and a sudden readjust- 
 ment may result in appreciable effects. A moderately 
 rapid change of the rotational velocity, by a fraction of a 
 second, would result in tangential stresses which would 
 throw up long north-south ridges (mountain chains). A 
 moderately rapid change of the ellipticity of the earth 
 would result in the throwing up of east- west ridges, and as 
 these two effects might occur simultaneously, we may have 
 diagonal ridges. Owing to the plasticity of the earth, it 
 keeps its surfaces, both interior and exterior, very nearly in 
 an equipotential condition at all times. Deviations may 
 accumulate for a short time and then be corrected sud- 
 denly (catastrophically ) , but the divergence is never wide. 
 The existence of ridges in the cardinal directions and their 
 diagonals, strikingly exhibited upon the earth's surface, 
 points to the dynamical (rotational) causes which we have 
 just considered. 
 
 During the times in the past also, when the axis was ± 
 to the ecliptic, the earth must have been subjected to 
 peculiar and violent stresses. At this point the angular 
 velocity about the 4 axis is a maximum, and this axis 
 shifts suddenly in the plane of the equator to a point 180° 
 opposite and then back again just before and just after 
 the pole is passed. Or in a comparatively short time the 
 
106 NATURAL PHILOSOPHY 
 
 comparatively large 1} velocity is reversed. Such a catas- 
 trophic commotion must result in extensive fracturing of 
 the earth's crust and the throwing up of east- west ridges. 
 
 The older ideas that the inequalities of the earth's 
 surface were due to the adaptation of its crust to a slowly 
 contracting (cooling) core, are found upon examination to 
 be untenable. The effect would be too slight to produce 
 the observed phenomena, and if there were such an effect 
 it would be entirely different. 
 
 We may therefore provisionally conclude that seismic 
 disturbances at the present time in all probability have 
 their origin in the earth's precession, and that the major 
 upheavals of the past probably were caused by minute 
 though rapid changes in the rotational velocity, and 
 especially at a particular time when the earth's axis was ± 
 to the ecliptic. There was probably a connection between 
 some of these upheavals and former glacial periods. 
 
 38. Sufficiency of Natural Forces 
 
 dv 
 Starting with the fundamental law, f = nt —7- , we 
 
 have derived all the main principles of natural philosophy 
 and explained many of the actions continually taking 
 place about us. The mathematical, or inductive, method 
 employed is merely a system of close and careful reasoning 
 — the only one by which absolutely true results can be 
 secured. Some of the proofs have been given in words 
 but are none the less mathematical for that reason. 
 Symbols are merely a shorthand for recording various 
 steps in the reasoning. 
 
 Natural philosophy is not limited to the scrutiny of 
 particular problems, but should supply us with an insight 
 into all matters — even the highest. It does not follow 
 that all or any of these higher problems will necessarily 
 ever be solved, but if they ever are solved it must be by 
 this method of strict and careful reasoning, or "organized 
 common sense." In all our experience we have never been 
 
SUFFICIENCY OF NATURAL FORCES 107 
 
 able to recognize more than two things, viz., matter 
 and motion, or simply moving matter. 
 
 The question arises whether there is something more 
 which we have hitherto failed to recognize — a tertium 
 quid. We know that life consists of moving matter and 
 that when the motion ceases the system becomes dis- 
 associated. Was there here something more? If there was, 
 it was not matter and it was not force or motion, since 
 motion exists only in connection with matter. 
 
 We have solved completely only a few of the simpler 
 phenomena resulting from matter in motion. As we ad- 
 vance in this study we notice the increasing complexity of 
 the phenomena. As the factors increase the results become 
 bewildering and our brains which are composed of only a 
 limited number of cells of matter in motion are unable to 
 follow them. Reasoning inductively from the complicated 
 phenomena which a few particles in motion can produce to 
 what an infinity of such particles under an infinitude of 
 forces should be able to produce, we can impose no limit 
 to the resulting phenomena. We have no right, therefore, 
 from our experience, to deny that there are any phenomena 
 which cannot be effected by matter in motion. We have 
 explored only a part of the field and beyond are vast 
 regions which may always be beyond our reach, but as we 
 progress there is nowhere any evidence of a limit — only 
 an unlimited vista and increasing complexity. The mathe- 
 matical theory of probability projects a certain trend into 
 the unknown, often with surprising accuracy, and both 
 our experience and probability point strongly to the 
 sufficiency of matter in motion. 
 
 The tendency of mankind, on seeing phenomena which 
 it does not understand, is to ascribe them to supernatural 
 agencies, and this attitude has an important bearing on our 
 present inquiry. Primitive men see in all the phenomena of 
 nature, spirits, good and bad. In the past, as in the pre- 
 sent, there have always existed in the minds of men hosts of 
 elves, goblins, ghosts, daemons and what-not, who are 
 
108 NATURAL PHILOSOPHY 
 
 continually performing supernatural acts. These spirits 
 and their acts have formed the bases of their religions and 
 to deny that our present religions are evolved from them is 
 simply to deny that there has been mental as well as 
 physical evolution. 
 
 When a Samoan is photographed he believes that a 
 daemon is in the camera and when a Hottentot hears a 
 phonograph he has no doubt but that a spirit is producing 
 the sounds. We have a smug way of imagining ourselves 
 very much superior to populations of the past, but the 
 difference is only in degree and very slight at that. We 
 have very recently learned how to utilize a few of the 
 forces of nature, which the average man does not under- 
 stand, but on the whole it is very likely that the average 
 ancient Egyptian was about as intelligent as the average 
 man of today. If an advanced intellect at some time in the 
 future shall look back upon both of us, he will probably 
 find little difference. For the most civilized and educated 
 among us firmly believe in miracles in the past, if not in the 
 present, and many of our ideas, if analyzed by such an 
 advanced intellect, would appear most extraordinary. 
 
 The average civilized man of today is in advance of the 
 Samoan in that he does not see the necessity of having a 
 daemon in a camera, but when it comes to a monad they are 
 on all fours and both insist upon the daemon. The natural 
 philosopher, with a fuller understanding of matter and 
 motion cannot share this belief. And he has not the least 
 desire that others should share his view unless they can 
 recognize that the probability, amounting almost to a 
 certainty, is that the camera and the monad are both 
 phenomena of matter in motion. It is not impossible that 
 we shall sometime be able to produce life artificially. 
 
 The vexed question of the immortality of the soul is a 
 simple one for the natural philosopher. Under all argu- 
 ments for such an immortality there stands out plainly the 
 personal desire of the pleader that, having existed for 
 some few years as a congeries of certain moving carbon, 
 
SUFFICIENCY OF NATURAL FORCES 109 
 
 nitrogen, oxygen and hydrogen atoms, he may somehow 
 and in some way continue a very different kind of existence 
 for all eternity. While all the rest of nature is continually 
 undergoing flux and evolution, he alone remains fixed 
 forever! There have been many who have had no desire 
 for such an eternal existence, but they have based no 
 arguments upon their personal wishes. Both the matter 
 and the motion of a living organism are immortal, but they 
 no longer form the same system after dissolution. The 
 natural philosopher cannot hold the view that the billions 
 of the earth's past populations — without considering the 
 lower animals — still exist as separate entities, or dis- 
 embodied souls. 
 
 It may seem that such questions are wholly foreign to 
 our subject, but the domain of the natural philosopher is 
 the whole universe, and there is nothing in it he may not 
 philosophize about, provided he preserves the strict 
 methods of his science. To many, such questions may seem 
 to be axioms, unworthy of serious discussion. But it must 
 be remembered that the great majority of living men 
 firmly believe in miracles and are convinced that the 
 laws of nature are not really laws, or at best are only laws 
 for a part of the time, since occasionally they are broken. 
 It is possible that in some higher stage of advancement 
 mankind may finally use the means at his disposal for 
 obtaining the truth and cleave to it. 
 
NOTE 
 On the Cause of Gravitation 
 
 The ether is the seat of an enormous store of energy as 
 evidenced by its enormous pressure. In the last analysis 
 this energy must be kinetic, or due to some kind of motion 
 within the ether, although we have as yet not the slightest 
 conception of the nature of such a motion. The atoms of 
 gross matter, being imbedded in the ether, necessarily 
 partake of this motion just as specks within a liquid par- 
 take of the motions of the surrounding molecules, con- 
 stituting the well known Brownian movements. They are 
 thus foci which reflect and radiate the internal ethereal 
 vibrations. 
 
 The atoms being in incessant motion and the ether 
 possessing both elasticity and inertia, among other dis- 
 turbances, longitudinal waves necessarily result. 
 
 We shall prove that such longitudinal waves necessarily 
 cause an attractional action between all atoms of gross 
 matter. A longitudinal wave is composed of two halves 
 having opposite properties. In one half the medium is 
 above its normal density and its particles are moving with 
 or against the wave direction, while in the other half the 
 medium is below its normal density and the particles 
 are moving in a reversed direction. An atom swept by the 
 wave will therefore be urged alternately towards and away 
 from the radiating point. 
 
 Let V be the average velocity of the stream in either 
 direction, a the amplitude of the wave, or the maximum 
 distance any particle of the medium moves from its 
 position of equilibrium, V the wave velocity, D the density 
 of the medium, P its pressure, I a wave length, and t the 
 time of a complete vibration. The force with which such a 
 
 110 
 
ON THE CAUSE OF GRAVITATION . Ill 
 
 stream urges an atom is proportional to its velocity, to its 
 density and to the surface which the atom opposes to the 
 stream. Taking the cross section of the atom as unity, 
 its mass as M, and the mass of an equal volume of the 
 medium as w, it is evident that such a stream striking M 
 and m at rest, will move them, in the time of half a wave, 
 distances inversely as their masses, or while it moves m a 
 distance 2a, just as it moves any other portion of the 
 
 medium, it moves M only — rj- . Or Ms = 2am, where 5 is 
 
 the distance M is moved during a half wave. At the end of 
 a half wave, therefore, M is 2 a -^^ — r^^ — behind m. 
 
 The time taken by each half wave to clear m is ^ , but 
 the time taken by the compressed half to clear M is 
 
 2V "^y MV / 
 while the time taken by the expanded half to clear M is 
 
 2V+ ^"[-M^^) 
 
 The average stream pressure in either direction is the 
 same and equal to kDv, k depending upon the units chosen. 
 But the time during which this pressure acts is unequal 
 in the two halves. There is a net pressure acting for a 
 
 time 4 a I ^ j, in the direction of motion of the ex- 
 panded half, during the passage of every complete wave. 
 This is equivalent to a force acting continuously equal to 
 
 kDv . /M — m\ T ,, .^ ^. , ^, 
 
 — — . 4 a I yr I . In all gravitational waves the 
 
 expanded half moves towards the radiating source. 
 I 
 
 v 4 a 
 Since jz = -y- , the force urging the atom contrary to the 
 
 wave direction is kD — j— . — r-r- — . The potential and 
 
112 NATURAL PHILOSOPHY 
 
 kinetic energies in a complete wave are equal, and it can 
 be shown (v. "Mechanics of Electricity") that the total 
 energy in a wave, per unit cross section, is 
 16 a2 DV2 _ 16 a2 P 
 I ~ I ' 
 
 Calling this total wave energy, E, the attractive force 
 
 ^ ~ V2' M ' t' 
 
 Now — is the energy crossing unit surface in unit time, 
 
 or the flux of energy per unit surface. We may call it 
 
 the density of the energy flux. The term "Flux of force," 
 
 frequently used, is meaningless. There is no such thing 
 
 as a flux of force, but a flux of energy constitutes a force. 
 
 If we take an equal volume of the medium in place of M, 
 
 or make M = w, there is no force and the atom merely 
 
 vibrates about its position of equilibrium. If the particle is 
 
 less dense than the medium, the force becomes negative 
 
 and there is a repellant instead of an attractive action. 
 
 It is evident that a body less dense than the ether will be 
 
 driven up very quickly to the limiting velocity, V, when, 
 
 since it travels with the wave, all further action ceases. 
 
 If M is very much denser than the ether, as is the case for 
 
 k E 
 all gross matter,/ = 77- . — , or for all gross matter oppos- 
 
 ing unit surface to the wave, the attraction is simply pro- 
 portional to the density of the energy flux. 
 
 Considering positive and negative charges of electricity 
 as differentiated portions of the ether which are respec- 
 tively denser and less dense than the normal ether, elec- 
 trostatic attractions and repulsions necessarily result from 
 the causes just discussed {v. "Mechanics of Electricity"). 
 At a time when longitudinal waves in the ether were 
 denied, as in fact they are today, Lord Kelvin wrote, 
 "I affirm, not as a matter of religious faith, but as a matter 
 of strong scientific probability, that such waves (com- 
 pressional) exist, and that the velocity of this unknown 
 
ON THE CAUSE OF GRAVITATION 113 
 
 condensational wave is the velocity of the propagation of 
 electrostatic force." Lord Kelvin, Baltimore Lectures. 
 
 There is no doubt that all atoms ceaselessly radiate 
 longitudinal waves. Such being the case, universal at- 
 tractions and repulsions necessarily follow. Hence, if any- 
 one should seek to explain gravitation through some other 
 agency, he would still have this agency unavoidably 
 coupled with it. Nature works in the simplest way possible. 
 She does not employ multiple agencies to produce a simple 
 effect. Further there is no other conceivable agency by 
 which such an action could be effected. 
 
 This action of longitudinal waves is not confined to the 
 ether but is a property of longitudinal waves in any 
 medium. It is readily verified experimentally in air 
 (sound) waves. Gravitational force is the push, not pull, 
 of the ether streams against the atomic surfaces, and hence 
 is proportional to the cross section opposed. The work of 
 some experimenters seems to show that certain atoms, 
 or at least certain arrangements of atoms (molecules) may 
 have different cross sections in different directions. Thus 
 Heydweiler claims that the weight of a crystal of CuSO^, 
 where the atoms are presumably oriented, is not the same 
 as that of the same mass in solution, where the atoms are 
 supposed to be unoriented. Wallace claims that the 
 weight of a given mass of water changes after it is frozen, 
 i.e., crystallized or oriented, but as yet we have very little 
 knowledge concerning such matters. The weight of a given 
 mass should vary with the orientation of its atoms to the 
 field, if the cross-sections of the atoms vary with the 
 direction. 
 
 END 
 
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