V IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I I^|2j8 |2.5 1.8 Hiotographic Sciences Corporalioii 4 // .^/ ^«?^% C ented by the square bf the velocity divided by the radius. Let V bo the velocity, r the radius, / the force which acts towards the centre. Let t be the small time in which the body, not acted upon by the central force, would describe the small portion A.D. of the tangent ; and let D.B, be the deflection by which the body is brought to B. Hence, at the limit, A.B. = V. t. B.D. = J ft\ But if A. E. be the diameter, the triangles E.A.B., ^ ,ii.D., are similar ; for the angles A.D.B., A.B.E., are right angles, and E.A.B,, A.B.D., ai*e equal. Hence E.A. : A.B. :: A.B. : 5.Z).; therefore J?. A X B.D = A.B. X A.B. ; and at the limit, B.D. = /<^ A.B. = v.t. ; hence 2 ;• x J/il- (v. t,y ....Therefore /. = r^" ! I This explanation (or demonstration) contains the re- markable substitution (as above) of A.B. for A.D., an assumption apparently that A.B. and A.D. are equal. Since it is 'obvious on mere inspection that A.B. is greater than A.D., the idea may suggest itself that thi.:^ CENTRIFUGAL FORCE. 11 substitution is an oversight or misprint ; but on examin- ation it appears to be an error of a different kind ; viz., it appears that the statement has been made under an impression that A.D. compounded with D.B., results in A.B., (i.e., that A. arrives at £. in the same time in vvliich it would, if not compounded with D.B., have arrived at D). But not even a reference to such a pro- position (as demonstrated) could justify in this place the substitution of A.B. for A.D., since here the object of the reasoning and the illustration is particularly to ascer- tain and demonstrate the value of A.B., anc^ its relation to that of A.D., and of other quantities. Therefore, to define precisely our objection.... vve agree that if the chord A.B. of the arc (in ^^'hewell's illustration) be taken as the velocity, D.B. will coi-rectly represent the deflection as a magnitude of force, but we object to the assumjition that the body moving with a velocity which would carry it tangentially to D. in the unit of time, will, when deflected, be enabled by it to arrive at B. in the arc ; and we also object that if the actual velocity of a body moving in a circle be taken, the rule will not apply with correctness unless the velocity be reduced in the ratio of the length of the chord to that of the arc. To avoid misunderstanding it may be observed that tlie above cal- culation would be in itself correct, if A.B., a part of the circle, were taken in the first instance as the space moved through in the definite time (denoted by t.). But A.D., representing the tangential motion from the point A-, m\ist be made equal to A.B., the arc ; and conse- quently D.B. will not be a (perpendicular) straight line but a curve, and will b jre^. " than D.B. shown in the figure. We wish here to draw particular attention to the important circumstance, which we shall presently have to consider uiore particularly, that in the propo* 1' 12 CENTRIFUGAL FORCS. sitioii here laid down by Whewell, the arc A.B., and not the (chord) straight line A.B., is the base of the triangle A.E.B. ; consequently if the triangle B.A.D. is considered similar to A.E.B., D.B., the base oi B.A.D. , must be also a curve (arc) and not a straight line. The rule and the exposition of the case given by Whewell, take the effect and from that derive a single impulse sufficient to produce the effect , now, in fact, it is not a single impulse which operates and results in that effect, but is a progressiva deflection occasioned by a continuous succession of impulses. To show that the unsupported assumption is not pecu- liar to Whewell's treatise but is at the present time a part of the recognized scientific teaching on the subject, we refer to Lardner's Mechanical Philosophy, Page 147, figure 63 : " Let P. be the fixed point to which the string is attach- ed. Let A. be the ball, and let A.C.F. be the circle in which the ball is whirled round. Lot A. C. be small arc of this circle moved over in a given interval of time. Starting from A., the motion of the ball has the direction of the tan- gent A.D. to the circle, and it would move from ^. to Z>. in the given interval of time, if it were not deflected from the rectilinear course ; but it is deflected into the diagonal A.C, and this diagonal, by; the composition of forces, is equiva- lent to two forces represented by the sides A.D., A.B. But the motion A.D. is that which the body would have in virtue of its inertia; and therefore the force A.B., directed towards the point P., is that which is impressed upon it by the tension of the string, and which, combined with the motion A.D., causes it to move in the diagonal A.C" I CENTRIFUGAL FORCE. 13 Here we find the same unsupported assumption in a somewhat different form. The arc A.C. is taken as representing the space moved over in a given interval of time by A., which is attached by a string to the central ])oint P. ; it is then stated that if the motion of A. were not deflected by the attachment to the central point, A would move (i.e., would have moved) in the same inter- val of time to B. But A.B. is less than A.C, therefore the velocity (in the deflected movement to C.) haj been increased, and this increase in the velocity is distinctly attributed to the tension of the string causing a motion in the' direction of the centre P., and this motion repre- sented by A.B. is assumed to compound itself with the motion A.B. and to result in A.C. If it be granted for a moment that such assumption may be true, it will immediately follow that the case must be one of uni- formly accelerated motion increasing from A. through- out every divisional part of A.C, and the velocity of A., when it arrives at C, must be accordingly greater than when it passed the point at A., and so continue to increase throughout the circle, {i.e., throughout every part of each successive revolution). This obvious cor- ollary is apparently quite overlooked. To substantiate tlie correctness of this teacliing, or, in other words, to demonstrate the conclusion thus arrived at by Whewell and Lard- ner, it would be necessary to adopt as a postulate, or to demonstrate in the first instance, that if a force act continuously on a moving body at right angles to the direction in which the body is moving, such force, by compounding itself in its effect with the motion of the body, produces accelerated motion in the body (increased velocity). Where is demonstration on this point to be found ? Has any one even ven- tured directly and positively to assert such a proposition ? i 1 ( 14 CENTRIFUGAL FORCE. There are three ways differing from each other, in which the fact that the motion in the circle is not accelerated may be explained : (1.) It may be assumed* that the t'vo motion.", the centripetal and tjie tangential, are only apparently combined, but strictly speaking, continue distinct, and that therefore the motion in the arc is a com- pound resultant continually reproduced, and not a single resultant motion ; consequently the motion in the arc contains the velocity in the tangential direction, but contains, also, independently, the velocity in the dirtction perpendicular to the tangent; therefore the velocity in the tangential direction is less than the velocity in the arc, and the body, if the centripetal force suddenly discontinued to act, would pro- ceed in the tangential direction with a velocity less than the velocity of the orbital motion in the circle. By th's assumption (which, however, we opine, is quite untenable,) the illustrations of Whewell and Lardner would become almost correct ; since A.D. would then represent the actual distance in the tangential direction through which the body would move in the unit of time, and the velocity A.B. in the ars would be greater because compounded with the deflecting force. Nevertheless, even on this assumption, the deflect- ing force D.B. must be (correctly) represented by a curve and not by a straight line. (2.) It may be assumed that at the point A., th'j tangential impulse is operative in such wise that the centripetal force has to overcome and deflect an actual tangential motion ; now such assumption in- volves as a corollary that the force acts at a certain very limited angle, greater than a right angle, in restraining! and deflecting the motion. For so long as it acts at a right angle only, there is evi- dently no actual tangential motion, and there can be no restraint. But if it be assumed that the force acts thus obliqueiy to the tangent in however small a degree, it is manifest that a part of the force will directly oppose and retard the motion, whilst another part thereof will directly aid and accelerate the motion. By this assumption, therefore, the centripetal force is resolved into a deflecting force, a retarding force and an accelerating force; and the two last being equal, the actual velocity of the moving body is neither accelerated nor retarded. • This is merely to explain a view which seems to be sometimes held ; such asBumption would be certainly false. * CENTRIFUGAL FORCE. 15 (3.) It may be assumed that the centripetal force acts as a restrain- ing force (at the point A. for instance,) always strictly at right angles to the tangential motion, and, as it at each moment counterbalances the tendency of the latter to increase the distance of the moving body from the centre, it therefore opposes and overcomes a true centrifugal force, and does not in any degree oppose or retard the actual motion of the body which is at right angles to the direction of the active force. At first it might seem that opposition and restraint necessarily involve some retardation of the motion, but it must be remembered that the action and reaction are equal, and as the force operates at right angles to the motion, there is no cause of retardation more than of accele- ration. It might also appear at first, that a body restrained in a circular orbit by the influence of a central gravitating mass, is not in the same case as a botly attached to the centre of revolution by an inelastic line, but in the latter case if the line be attached to a weight free to move, i\nd sufficient to counterbalance the centrifugal force, as in the whirling-table, the effect is the same and the conditions become strict'.y similar to those of the body restrained by the gravitation of a central mass. Of these three explanations and the assumptions to which tliey respectively belong, we opine that the last is that which is prefer- able, because most probably the strictly correct and sound explana- tion of the case ; but, if it be so, the present teaching, as shown by our quotation, involves a fallacy, namely, that the tangential velocity, or impulse, A.D., is less than that of the arc A.B., and that the deflecting motion or force D.B. is a straight line, whereas it is necessarily a curve.* • We. have elsewhere called this curve the arc of deflection. See page 24- {2) Terrestrial Gravitation at the distance of the moon. A practical application of the proposition stated in the foregoing quotation from Dr. WhewelFs treatise is to be found in the method of comparing the force of teiTcstrial gravitation at the distance of the moon with that exerted on a body close to the surface of the eai'th. Lardner's Astronomy. 2614. " Method of calculating the central force hit the velocity and curvature. — Now the space through which any central attraction would draw a body in a given time can be easily calculated, if the body in question moves in a circular or nearly circular orbit round such a centre, as all the planets and satel- lites do. 3! * \ Let E., Fig. 745, be the centre of attraction, and E.m. the distance or radius vector. Let m.m.' — F., the linear velocity. Let m.n. and m/n-. be drawn at right angles to E.r,i., and there- fore parallel to each other. The velocity m.m.' may be considered as compounded of two, one in the direction m.n.' of the tangent, and the other m.n. directed towards the centre of attraction E. Now if the body were deprived of its tangential motion m.n.', it would be attracted towards the centre E., through the space m.n., in the unit of time. By means of this space, therefore, the force which the central attraction exerts at m. can be brought into direct comparison with the force which terrestrial gravity exerts at the surface of the earth." TERRESTRIAL GRAVITATION ON THE MOON. 17 '•'• It follows, tlierefure, that if /. express the space through which such body would be drawn in the unit of time, tailing freely towards the centre of attraction, we shall have f,—m.n. But by the elcnien- tary principles of geometry, m.n. X 2 E.m. = nun' ^ Therefore, / = — ' 2r that is, the space through which a body would be drawn towards the centre of attraction, if deprived of its orbital motion, in the unit of time, is found by dividing the square of the linear orbital velocity by twice its distance from the centre of attraction. r X Since a 20020-5, we shall also have / = r y. a 2 X 200205 The attractive force, or, what is the same, the space through which the revolving body would be drawn towards the centre in the unit of time, can, therefore, be always computed by these formulae, when its distance from the centre of attraction and its linearor angular velocity are known. 1290000 Since (2'5('3) a = this, being substituted for a in the preceding formula, will give /= 412541 X by which the attractive force may always be calculated when the distance and period of the revolving body are known. 261 5. Law of gravitation shown in the case of the moon. — The attrac- tion exerted by the earth, a;, Hs surface, may be compared with the attraction it exerts on the mooi. . by these formulte. In tlie case of the moon, V = 0-6.S56 miles, and r = 239,000 miles j, and by calculation from these data, we find /= 0-0000008459 miles = 0'05.36 inch. The attraction exerted by the earth at the moon's distance would, therefore, cause a body to fall through 536 ten -thousandths of an mch, while at the earth's surface it would fall through 193 inches. The intensity of the earth's attraction on the moon is, therefore, Jess than its attraction on a body at the surface, in the ratio of" 18 ORAVITATION AT EARTIl'S SURFACE. 1,930,000 to 536, or 3000 to 1, or, wliat is tlie same, as the square of GO to 1. But it lias been siiown tliat the moon's distance from the earth's centre is 60 times tlie eartli's radius. It appears, therefore, tliat in this case the attraction of tlie earth decreases as the square of the distance from the attracting centre increases; and that, conse- quently, the same law of gravitation prevails as in the elliptic orbit ot a planet." (3) Centripetal and eqitilihriaff centrifugal force of re- volution at the surface of the earth. Now from this application of the proposition stated by "Whewell the actual centripetal force at the earth's sur- face may be obtained from the velocity and centrifugal force of the moon. The space here given as fallen through in a second of time by a body near the earth's surface, viz., IG,^/ feet, is, strictly speaking, correct or incorrect accordingly, as whether the locality at which the faUing body approaches the earth's centre be near the equator, in one of the temperate zones, or near one of the poles of the eartli. For let the body be supposed to fall near the equator ; let E.F. (fig. 4)* be the surface of the earth which is rotating on its axis with a velocity of about 14G7 feet in a second, and let the body fall from the top of a high tower, at the point A., above the earth. Since the tendency of the body's motion is in the tangen- tial direction A.D., it will descend, not vertically to- wards the centre of the earth, but in the curve of a •parabola towards the point F. / it is true that ii will fall upon the point E. which is vertically beneath the point A., but whilst the body is falling, the point E. is advancing towards F., with the same velocity where- with the falling body advances. Now it is evident that were there no attractive force acting on the body from * Page 25. GRAVITATION AT EARTH's 8UUI ACE. 19 the earth's centre it would move along the line A.D. and increase its distance from the earth ; and, again, were the attractive force sufficient only to deflect the mo- tion of the body so .iiuch as to cause it to move through the arc A.B., the body would not approach any nearer to the earth ; therefore, supposing the body is found by actual experiment to fall through 16,'j-feet in a second in that locality, the total deflection from the tangential motion in a second is greater than IGVj ft. by the de- flection of the tangential line A.D. into the arc A.B. Let us see how much this addition will amount to. The time in which it would be necessary for a body to make a complete revolution around the earth, near to the earth's surface, in order that the centrifugal force shall be sufficient to counterbalance the attractive force of gravitation and enable the body to retain its distance from the earth, may be readily obtained from the actual velocity of the moon in her orbit. For the moon's dis- tance is known to be equal to 60 semi-diameters of the earth, the time occupied by the moon in completing a revolution is 655 hours, the intensity of the terrestrial attractive force at the moon is the 3600th part of the intensity at the earth's surface ; therefore, in the first place, the intensity of the force at the earth's surface being greater by the square of sixty times, this increased force will correspond to or equilihriate an angular velo city increased to sixty times that of the moon ; but, since the radial distance from the earth's centre has been dim- inished to the one-sixtieth, the centrifugal force, which is proportional thereto, is also diminished to the one- sixtieth and which is equivalent to a further augmenta- tion of sixty times the attractive force of the earth, and "20 - OBAVITATION AT EAaTU'8 HLRKACE. therefore equivalent to a further incrense of V()0 tiums ^ 7.8 times the angular velocity of the body moving in orbital revolution near the eartii's surface. We have, ' herefore, first, (JoO hours (livi(lei)roaching nearer to the earth's cen- tre, and the centrifugal force developed by this velocity is equal to an opposing centripetal attractive force of IGij ft. in one second, because such a velocity is equal to 4.95 of a mile moved through in a second, and the square of this quantity divided by tiie diameter of the earth, which measures 8000 miles, gives 193 inches. Now this velocity includes the velocity of the earth's rotation, which taken separately, is equal to 1407 ft. in a second, which quantity sqjiared and divided by the earth's diameter, equals about 0*0 of an inch. So that the actual space through which a body will fall in a second near the ec^uator is IGi^-Z^r = about 10^^ ft.* A difference which is equivalent to about 180 feet lees space fallen through in a minute at the equator than in the polar regions, therefore 57,900 ft. - ISO = 57,720 ft. in a minute. * By pendulum experiments the amount of difTerencc has been deter- mined at 1 part in 193, wliich would equal 1 inch in the 16i'j ft. The diff. found as stated, appears at about 1 part in 380, about one half as much as the quantity ascertained by the pendulum experiments. 4. The Rotation of the earthy or other planei , <(s mo flifu'^nrj the injlitcnce of the herc. i h Thoro is an iipparent ilin'uMilty of ii i»eculiar character when the mind Is first brought to the consideration of this stdjject, which may l.»»3 tlms stated : Let Fig. 3 rciiresent half of tlie earth's spliere; E.E.E. being a section at the e(|uator, and P. tlie jdace of one of the poles. The earth is rotating in the direction of the arrows, and a body on the surface at E. is therefon; revolving with that surface n found the centre of the earth at the rate of about 1,000 miles an hour. A body on the surface at or near P., is comparatively uninfluenced by this rota- tory motion. An argument to the following effect is therefore likely to suggest itself to the student : (I) A body revolving around a centre of gravitation (or central point) at the rate of a thousand miles an hour, must develop a very considerable centrifugal force. (2) This centrifugal force opposes the attractive force from the centre, and must therefore (3) be a deduction from the apparent gravity or weight of the body. Hence (query), is the (apparent) weight of the same body considerably greater in the polar regions than near the equator ? or is there any sensible difTerence in the (apparent) weight of the same body if removed from the one to the other situation ? The answer to this question is that small n UOTATION OF THE (JHAVITATINd MAMfl. ijiiiiiitity ordilli'iviic*! wliicli wc liiivo just been coiiNider- iiig, iiiiiiicly, of 1 part in ll):i luuiorclhig to peinliiliiiM ♦'XjK'riiiientH, iiiul of I part in about .'J'^O uccording to tlie more dirt'ct niotliod explained in the foregoing. But the (piantity of centrifugal force thus shown to bo actually developeventeentli the velocity, liave at least some considerable influence in reducing tlie eflect of the ciMitral gravitating force ? To explain the circumstance that in fact such influ- ence is very much less consideraljle tium it at first seems reasonable to expect, it is only necessary that the rela- tion of the centrifugal force to the angular velocity sliould be distinctly apprehended, namely, that the cen- trifugal force increases not simply as the angular velocity but as the scjuare of the angular velocity, so that an iricrease of twenty times in the velocity of the earth's rotation (in which case the earth would make twenty rotations in the same time it now takes to make one) would increase the centrifugal force not simply twenty times but four hundred times. (•) This is, of course, discarding tlie cartli's atmosphere ;—».«., supposing the earth to be without an atmosphere, because the air would evidently -resist and impede the motion of the body. 24 THE ARC OP DEFLECTION. The question as to whether a boJy revolving near the earth's surface can escape contact and continue to revolve, is evidently dependent upon equality between the versed sine of the arc through which the body moves in a defi- nite time and the distance through which gravitation would move the body towards the centre in the same definite time ; because if the latter measures the fonner, and, being equal, causes no more than the amount of deflection from the tangential direction which is required to produce the arc, the body will in that case continue to revolve, but if the deflection be in any degree greater tlian the versed sine of the arc,* then must the body fall to the earth. By taking the equivalent to a thousand miles at the equator in a circle, the case may be illus- trated, — as in fig. 4, where the angular divisions 1, 2, • We have explained that the deflection V.B. must be represented by a f.urve which is longitudinally greater than the versed sine, but it is not to be inferred that it is greater in magnitude of force, the effect is here analogous to that of a force acting through a lever and thereby moving a lesser weight through a greater space. For the magnitudes of force A.l!. (the straight line) and JJ.B. (the curve) are dependent on the time during which the force of gravitation acts freely in each case, and as the times are equal so must the force-magnitudes be equal. Therefore the versed sme correctly gives the quantity of force exerted, but the space, through which it is exerted in deflecting the body, is the arc of deflection and is longitudinally greater than the versed sine. The true arc of deflection (which is of great interest) may be thus drawn. Let the radius A.O. equal !'". Take a point, on the line ^. A, distant from A 200452. With the point thus found as a centre and the distance £>., describe the arc D.B. The arc A.B. thus cut off equals in length the straight line A.D. {See " The Circle and Straight Line," page 24, Fig. 25.) CENTRIFUQAL FORCE AT EARTH 8 SURFACE. 25 3, 4, 5, 6, are each equivalent to about 1000 miles ; and tlie body A. near the surface of the earth is supposed to be moving in the tangential direction A.D., with a velo- city of about 286 miles in 1 minute (which would carry it through a space equal i-u the circumference of the Fig. 4. earth in about 1 liour and 24 minutes).* The effect of terrestrial gravitation acting freely on a body near the surface is Ki own to equal in effect an approach towards the centre (or space fallen through) of 57,900 feet in a mmute. The deflection at the limit B, taking A.B. equal to 2000 miles, will be found by multiplying * — The earth's circumference is here taken at 24,000 miles, but it is in fact (7,925 X 3,142 = 24,900) very nearly 25,000 miles, and hence for a body to revolve around the earth in a permanent orbit, about 88 minutes (1 hr. 28 m.) will be required NoTK. — Strictly speaking, \{ A.E. equals 10 miles.. then C.A. : C.E. :: 4010 : 4000. But the time required for the body to make a complete revolution must be increased in more than this proportion because the angular velocity will be diminished ; for otherwise the; deflecting force would be insufficient." 26 THE EAETU 8 ROTATION. 67j900 by the square of thenumber of times 286milesis contained in 2000 miles, which is seven, .thus : (7 x 7) X 57,900 = 537J miles, or (about) 540 miles, which represents a centripetal motive force from A. of IGrV feet in 1 second, = the versed sine A.E. (5.) The rffect of gravitation in sustaining the rotation of the earth. From the foregoing explanation it will be evi- dent that a difference exists between the conditions of bodies on the earth's surface at or near the equator, on the one hand, and at or near the poles, on the other; for whereas the gravitation (attractive force) of the body is in both localities directed towards the centre of the earth, the pressure ("gravity) of the body is, when near the poles, also directed towards the centre of the earth, but near the equator this is no longer the case, because the pressure of the body must have that direction in which the body would actually move if (supposing a portion of the earth's surface to be suddenly taken away) it were free to approach more nearly the earth's centre ; and, as we have seen, this m.ition would be, not in a direct line to the centre, but in a curve compounded of the tangen- tial motion (due to the earth's rotation) and the centri- petal motion. With the actual velocity of the earth's rotation, the deviation, which is necessarily in the direc- tion of the earth's rotatory motion, from the west towards the east, will be, if we take the tangential deflection roughly as half an inch in 16 feet, about 10^ miles on the earth's horizontal diameter. The angle of deflection, therefore, is less than can be made readily appreciable by illustration, but if we suppose the earth's velocity of rota- tion increased 10 times, the tangential deflection will become equal to about 1000 miles on the horizontal line miles is 7x7) which )f 16r'7 lation of 1 be evi- itions of itor, on ;her; for body is he earth, near the irth, but lause the in which (ortion of ) it were ; and, as lirect line e tangen- le centri- tie earth's the direc- st towards deflection \ miles on deflection, reciable by ity of rota- ection will izontal line THE earth's rotation. .27 of the earth's centre ; and it will become evident, as indicated in fig. 5, that the attractive force from 'the central part (or inner sphere) of the earth by combining with the tangential motion is utilized as a means of sus- taining the rotation of the earth.* It thus becomes apparent that the attractive force acts within the sphere of the rotating planet upon the matter composing that sphere (excepting that part which occu- pies the central region) in the same manner as it acts upon a body revolving around that sphere, outside the planet, only that where the gravitative force acts upon the matter within the sphere, instead of the centripetal force being wholly employed in deflecting tlie tangential motion, a large proportion thereof is employed in bind- ing together the aggregated matter into one mass. But the hnea of pressure should be curves. See the figure Cfis 5) repeated in the Appendix. The difference, is tliat the obliquity of the pres- sure to the tangential line at the surface is a little greater than here siiowzi* (6) THE LAW OF GRAVITATION AND CENTRIFUGAL FORCE, Let a planetary body, Fig. 1*, governed from a centre of gravitation at C.l, move, at the distance C} 3., from that centre, through the arc D.B.A., in a definite quantity of time denoted by T. Now let us suppose the same centre of gravitation removed to twice the distance from B., namely to the place C.2, and let the planetary body again move with the same linear velocity in the same quantity of time T., through an arc described with the centre C.2 and the radius (7.2 B., and having the same point of bisection B. This second arc will be the arc d.B.a.; namely, an arc having the same actual length as the first arc but belonging to a circle of twice the magni- tude, because the length of the radius vector has been duplicated. If we now take the tangential straight line F.B.E., equal in length to each of the arcs, touching the circle at the point B., and having B. for its point of bi- section, we obtain E.A., and E.a., measuring respec- tively the amount of centrifugal force which requires to be counteracted by the gravitative influence exerted from the centre, in the case of each arc, namely in the first from C. 1, and in the second from C.2 ; and since E.a. is the one-half of E.A., it follows that the effective gravitative influence required has been reduced to the one-half by the removal of the gravitating centre to twice the distance. But the affective influence of the mass has been reduced to the one-fourth, therefore the mass of the gravitating centre will have to be twice as great as at the lesser distance. Let us next suppose that, with the centre of gravita- tion at the same more distant place 0.2, the planetary Plates at the end of Book. .-^^s / s ,^ I / 1/ I// \ -4- —I II < «^ \, \ 1^ y and tjje s^qqq between the point 0. and ANGULAR DYNASIIOAL EQUILIURIUM. 88 the point at the extremity of the arc moved through by the planetary body in the time T., will equal tiie mth part of the space 0. b. (h) ANQULAIt DYNAMICAL EQUILIIJKIUM. (Case in whicli tlio gravitative iiilluence is dynaniicaliv counter- lialuiiced by the centrifugal furce, tlie angular velocity remaining un- altered at various distances of the I tody from its itrimnry or centre of gravitative influence. .the mass of the primary being proportional to the cube of the distance and the linear velocity of the planet pro- portional to the distance.) Fig. 2 (h). Let g. be a centre of gravitation, at the distance y.V. from the planetary body P., subject to its influence. Let P. move in tlie direction P.N., and let the velocity of its motion be so proportioned to the force of gravitation by which it is restrained that it will move tlirough the arc P.B. and arrive at the point B. in the same definite time T. in wliich it would have arrived at the point TIf. if unin- fluenced by gravitation. M.B., therefore, will measme the centrifugal force counteracted by the force of gravita- tion. « Let the centre of gravitation be now removed to twice the distance from P., and let it be required that the planetary body shall move through the same angle in the same definite time T. as before (it will tlierefore ne- cessarily move with twice the linear velocity, because the radius vector being increased to twice the length the similar arc will be of twice the magnitude.) What increase in the mass of the centre of gravitation g. will be necessary in order to restram the planetary body from deviating out of tlie arc of the circle ? 84 ANUULAR DYNAMICAL EQUILIBRIUM. Since the arc P.E., tlirough which P., will now move is similar to the arc P.H., und twice tho magnitude of IMi., the Np«C(j E.N., measuring the centrifugal force which will now re(|uire to he counteracted, \n twice a« great as the space H.M., and, therefore, (since this is to he accomplished in the same time T.,) twice the amount ofeH'ectivegravitative intluencewillbo required to operate u[)on P., from the centre of gravitation in the same time. But, by the law of gravitation, because the distance of g. from P., has been doubled its ettective influence has been redu(;ed to the one-fourth, consequently, therefore, the mass of ^. must be increased twice four-fold ; the centre of gravitation will be accordingly represented by 8 g. ami its effective influence at P. by 8 divided by 4 , — that is, an amount of influence twice as great as when the lesser mass acted from half the distance. Again, let the centre of gravitation be now removed to four times the original distance from P., and let it be again req^^red that the planetary body shall move through the same jaigle in the same time I", as before (it will there- fore move with four times the linear velocity, because the radius vector is increased to four times the length.) What increase in the mass of the gravitating centre g. will now be necessary in order to restrain the planetary body from deviating out of the arc of the circle ? Since the arc P.e., through which P. will now move, is similar to the arc P.P., and of four times the magni- tude of P. B., the space e.n., measuring the centrifugal force which will now require to be counteracted, is four times as great as P.M., and, therefore, (since the time i» still the same) four times the intensity of gravitative influence will be required to operate from the centre of i s f ^ '■ •■+■- ' U/) .-.. ..(/, Fig 2. ■m^ STATICAL EQUILIBBIUM. 3& gravitation. But, by the law of gravitation, because the distance of g. from P. has been increased four-fold, its effective influence has been reduced to the one-sixteenth^ consequently, therefore, the mass of g. must be increased (4 + 16) sixty-four-fold ; the centre of gravitation will be accordingly represented by 64 g. and its effective influence at P. by 64 g. divided by 16, that is, an amount ot influence four times as great as when the lesser mass (g.) acted from one-fourth of the distance. Wherefore, generally : — if D. represents the distance g. P., and R. the linear velocity with which P. moves through a definite angle in the time T., then if D. be increased m. times and it be required that P. shall move through the same angle in the same time, the mass of ^r. must be increased m^ times, and the linear velocity of P. (at the distance Dm) will be 72 X m. ; that is, will be m. time? the linear velocity with which it moved at the distance 2). (c) THE DYNAMICAL CONDITIONS OP STATICAL EQUILIBRIUM. (Case in which a planetary body situated between them, is subjected to the equal opposing influence of two centres of gravitation, one of them being of greater mass than the other, and at a greater distance from the planetary body than the other. It is required to investigate the dynamical relation of the planetaiy body to each of the centres of gra- vitation respectively.) Fig. 2 (c.) Let L. be the lesser, and G. the greater centre of gra- vitation, and P. the planetary body. And let the dis- tance of G. from P. be four times as great as the distance of L. from P. ; then, by the Law of Gravitation, since the influences of both at P. are equal, the mass of G. is 36 STATICAL EQUILIBRIUM. necessarily sixteen times as great as that of L. If P. move at right angles to a line joining the two gravitating centres, since equally acted on by th'^ opposing influences of both, its motion will not be restrained by either, and it will therefore move in a straight line at right angles to the line joiiiing the centres. Let it be supposed that the greater mass G. be removed, and let the velocity of P's. motion be so proportioned to the influence of L. that P. will be constrained to move in the arc of a circle around L. and in a definite time T. will arrive at the point B. B.M. will accordingly measure the centrifu- gal force counteracted in the time T. Now let it be supposed that the lesser mass /,, is removed, and let P. move under the influence of G., in the same direction as before, with a velocity so proportioned to the gravitative influence of G. that P. will be constrained to move in the arc of a circle around G. Since the deflection of the planetary body from the tangential line in a definite time measures the effective gravitative force exerted in that time, and since the radius vector with which the arc is described has been now increased four-fold, it is evident that if the linear velocity be not increased the deviation at the extremity of the arc would be, as shown in case a, only the one half of P.ilf., therefore the linear velocity must be increased in order that the centrifugal force shall equal the gravitative influence. Now it has been already shown in case 6. (where N.d. = B.M.) that if the linear velocity in the arc of a circle be doubled whilst the distance of the centre of gravitation is quad- rupled, the deflection from the tangential line will be the same as before, therefore it is evident that, for the deflection to equal B.M. the linear velocity must .be increased to twice its former amount, and the arc P.d.f STATICAL EQUILIBRIUM. 8T twice the length of the arc P. B., will be the arc through which the planetary body P. will require to move in the time T. in cder that equilibrium between the centri- fugal and gravitative forces may restrain that body from deviating out the circle : consequently d.m.= JB.M., and the conditions of Dynamical Equilibrium are fulfilled. For, if this be not admitted, let it be supposed that the linear velocity of P. is not increased, then, evidently, P. cannot move in the arc of a circle around G. because the effective influence of G. at the distance G.P. is equal to the effective influence of L. at the distance L. P., and, therefore, if the velocity remn i as before the deviation from M. towards G. must equal the pre- vious deviation from M. towards L., and P. would move through a curve P.b. inside the arc P.D.d.* Consequently^ for example, if a solar planet be supposed to have less than the linear velocity of equilibrium, it would neces- sarily approach the sun (in an inwavd spiral or helical curve of decreasing radius) until the conditions of dyna- mical equilibrium were attained. That the attainment of the conditions would result from the a|>p roach is evident, for the increase in the angular velocity would compensate for the increase in the intensity of gravitative force at the diminished distance, as shown in case a., and the motion generated in the body by approaching (falling) • This rule or law is dependent on the relation of the curve which measures the deflection from the tangential line, to the radius (vector). This curve, which is of great interest and importance, may be termed the complementary arc of deflection. A notice of its relation and properties will be found in the appendix to Part Third of 'the Circle and Straight Line,'where we have termed it the comple- men tary arc of curvature. That part of the tangentialline cutoff' by the {c.a.d.) curve is in each case equalin length to the arc. ii I F. 38 STATICAL EQUILIBRIUM. towards the centre of gravitation would be an absolute addition to the velocity at the diminished distance. (Note, The curve P.h. will be a very little greater than the arc of a circle described with centre I. and radius I.P., and there- fore, also, a Utile greater than the arc P.B. : because the linear velocity oi the moving body would increase in consequence of the deviation inside the circle.) On the otlier hand, returning again to the lesser centre of gra- vitation L. at the distance L.P., and, discarding the greater centre G., let us suppose the body P. to move with that duplicated linear velocity with which when in- fluenced by the gravitative force of G. it described the arc P.d. in the time T. Here again the body P. will be unable to move in the arc of a circle, for B.M. measures the effective gravitative influence exertad by L. upon P. in the time T., and, therefore, P. will move in the curve P.e. outside the circle, and m.e. will be (very nearly) equal to m.d. Consequently, for example, if a solar planet be supposed to have a linear velocity exceeding that required by dynamical equilibrium, it would necessarily retire from the sun (in an outward spiral or helical curve of increasing radius) until the conditions of dyna- mical equilibrium were attained. (Note. The curve P.e. will be a very little less than P.d., because the velo- city will be diminished by the deviation of the moving body outside the arc of the circle.) Wherefore generally : If a planetary body situated between two centres of gravitation, and more distant from the one than from the other of them, be in statical equilibrium, (that is, influenced equally in the two op- posite directions,) the dynamical conditions of the planet- ary body relatively to each of the centres of gravitation < ; H 7 T :l A \ V \ \ ■ ^ ^ 'I STATICAL EQmunUIUM. 39 may be thus stated Let the diHtance of the greater from lie planetary body be 4 times the distance of the lesser (by the Law of Gravitation tlie mass of the greater will then be 10 times that of the h'sser), then if V. be the linear velocity of equilibrium with which the planetary body will revolve (in a circle) around the lesser, uninfluenced by the greater; 2 V. will be the linear velocity of equilibrium with which the planet will revolve around the greater, uninfluenced by the lesser. A statement of the general relation . . as a general law, would involve more complexity than is desirable in this place. A reference, however, to Fig. 3, will enable that relation to be at once appreciated. — The successive arcs, cut off by the same straight line a. b., represent the linear velocities of equilibrium belonging to the dynami- cal conditions of statical equilibrium in each instance ; the radius of each successive arc represents the relative distance of the primary from the planet, the mass of the primary being understood to be increased proportionally to the square of the distance.* * It thus becomea at once apparent that, comparing the primary of a planetary system, in reference to a Satellite of that system, with that of the great central primary, if the mass of the latter be supposed greater than that of the planet in proportion only to the squares of the distances, and the distance of the central primary be very great, although the inti isity of gravitative influence on the Satellite will be only the same from each, yet the linear space, throughout which that amount of force will be effective from the distant centre, will be considerably greater. Also, for example, if we suppose the distance between a planetary body and its prim- ary increased 100 times, and the ma^s of the primary lo be also increased 10,000 times, altliough the stnlical intensity of gravitation will be the same as before, yet that amount of attractive force will operate throughout a (4,- 47a to 1 ; the sun having a real diameter of 8Sa,()0() miles. The density of tlie viHihle sun, therefore, is less than that of the earth in about the proportion of 1*0() to ;3'() (0-27.) (8) I'r.ANKTAHV NYSTKMS. There is a circumstance of great importance in rela- tion to the general case under consideration, which, as it seems to us, has been in a measure overlooked l>y writers on astronomical subjects. It is that, when* a planet is attended by a satellite or by satellites, that entire planetary system, botli the primary and satellites, is in r if I 46 A PtANET APPROACHING THE SUN. In the first place let us remark (in opposition to what might sug- gest itself for the Tnoment as a reasonable inference) that the increase of velocity consequent upon such an approach of the body for any definite distance must be proportionally the same whether the boJy, being at a great distance approach, subject to a very feeble attraction, or, being at a much less distance, approach subject to a much greater intentity of attractive force. In the next place, we observe the fol- lowing correspondence in the ratios and relations of the elements belonging to the case. .(1) The ratio of the increase in the velocity of a falling boily according to the law of accelerated motion ; by which the space fallen through in a definite time is proiwrtional to the jiquare of the time, and, since equal divisions of orbit represent equal increments of time, also proportional to the square of the angular velocity. (2) The attraction of gravitation, which increases propor- tionally to the square of the decrease in the distance. 3) The ratios of the areas of circles ; because the areas of circles are to each other as the squares of the radii. The first pnrt of the question we liave to consider, is ..What will be the increase in the angular and linear velocity of the body ? This question, however, at once suggests the more general question, upon which it is evidently dependent, whether, supposing the approach to be a consequent of excess in the attractive force, there be a necessary and direct relation between the primary velocity (i.e., the velocity before the approach commences), the amount of the approach, and the in- creased orbital velocity resulting from that approach ? Now, since, if there were no excess in tlie gravitative force at the greater distance there would be no approach, it is evident that there is such relation: for (1) the amount of tiie approach under the influence of gravi- tation necessarily implies an excess of gravitative force at the greater distance proportional thereto ; and, also, (2) the primary velocity which measures that part of the attractive force counterbalanced by the centrifugal A PLANKT APPROACHING THE SUN. 47 force at that velocity, musji, be therefore related to the accelerated velocity ; because, for the velocity to becoMie equilibrially proportional to the entire force, the addi- tional velocity must be proportionally equal to the excess of the force, and the primary velocity must have accordingly a ratio to the accelerated velocity propor- tional to the ratio which that part of the force counter- balanced previously to the approacli, has to the whole of the force, which is counterbalanced after the approach has taken place ; for example, if the approach reduce the distance to the one-half, the excess of the force at the greater distance must have been equal to that part of the force employed in deflecting the tangential motion and counterbalancing the centrifugal force before the approach commenced, so that, if we call that p""t of the force employed before th3 approach commenced/., the whole of the force must be / x 2 = 2 /. This example also indicates the precise nature of the proportion between the whole force and the diminution of the distance ; for, calUng tiie primary distance D, 2/-D-2 = i)- (7)-f-2) {i.e., the double force reduces the distance to the one-half), f therefore / + -|- will equal D - {D f 4), and. -/ +-J will equal D - (D ^ 8), and thus, generally / + Z will equal D - (D -^ 2m.) m And since 4 /must equal Z) -^ 4, therefore, also, m/wiU equal D -^ m. Ij^^ 48 A PLANET APPROACHING THE SUN. By the law of gravitation the increase of the force i* as the square of the decrease in the distance, therefore 2/ at the greater distance, when the distance is reduced to the one-half, must equal in effective influence 8 /, i. €., 8 times the amount of the force which was employ- ed in controlling the motion at the greater distance. Now it is established in Mechanical Science, from ex- periments with the tvhirling-tahle, &c., that the centri- fugal force is proportional to the radial distance of the revolving body from the centre of revolution, therefore a diminution of the distance to the one-half is practically equivalent to doubling the active force, so that 8 x 2 = IG times the amount of the force which operated at the primary distance before the approach took place* As the body constantly moves in a curve of which the radius is continually diminishing in length, the approach towards the centre'is a part of the deflection by which the body is drawn from the tangential motion, and that part is the excess of the deflection over and above what is suflicient to cause the body to move in the arc of the circle*. But since the excess of force, in the first in- stance, equals that which is required to deflect the tangential motion into the arc, so must the entire deflec- tion or centripetal motion, when the approach has been completed, have effected an increase in the absolute vel- ocity equal to the linear velocity at the original distance ; and, since the distance has been then reduced to the one * At first it may appear that the attractive force will constantly increase in intensity as the body approaches nearer, but it must be remembered that such increase [as stated above] is constantly counterbalanced and neutral- ized by the increase in centrifugal force consequent on the increased angular velocity ; hence, this element of variableness does not rei lly belong to the conditions of the case. A PLANEX APPROACHINQ XBE SCN. 49 half, the angular velocity with which the body moved at the greater distance has been thereby doubled; hence, the total increase in the angular velocity of the body at the half distance, supposing the approach to have been caused by excess in the gravitative force, will be (2 x 2 * 4) four- fold ; and since a four-fold increase in angular velocity requires a sixteen-fold increase in gravitative force, dyna- mical equilibrium will be attained and the body will con- tinue to revolve in a permanent orbit at the half-distance. Moreover, it will follow that (1) since the areas of cir- cles are as the squares of the radii, the radius vector having been reduced to the one half and the angular velocity increased four-fold, the area swept by the r.'.dius vector of the body revolving at the shorter distance must equal the area swept by the radius vector of the body before the approach commenced. And, (2) because at the half- distance the area remains equal, it is at once evident that at all the intermediate distances greater than the half- distance the area must also, under the same conditions, remain undiminished. Let us apply this exposition to a particular case in theoretical astronomy, and suppose the angular velocity of the earth's orbital revolution has been by some sudden cause reduced from 1 revolution in 3Go days to 1 revolu- tion in 51G days, that is to 0-7071 revolutions in the same time, then, since 0*7071 x V2 = I'OO revolutions, only one-half the active force of the solar gravitation will be employed in the requisite deflection and the remaining half of the force will be in excess, therefore tlie earth will immediately commence to approatih the sun, and the approach will continue until the half distance has been reached ; the altered conditions will then be that the ^ J^n i <) I I" 60 ▲ PLANST AtPROACHINO XHK SON. earth will revolve with four times the angular velocity, completing 1 revolution in (OlG-^4) 129 days; to con- trol which the active force is prr,ctically IG times that which was employed at the greater distance, for only half the active force was then employed and the whole force has been increased four-fold by the approach to the Bun within the half-distance, and the centrifugal force has been reduced to the one-h.alf by the diminution of the radial distance from the sun to which it is propor- tional, therefore equilibrium is attained and the earth will continue to revolve in a permanent orbit at the half- distance with a velocity of 1 revolution in 129 days. Also, since the radius vector is now diminished to the one-half and the angular velocity is increased four-fold, the area now swept by the radius vector will equal the area swept by the undiminished radius vector at the former distance moving with the lesser velocity. On careful consideration it will become apparent that the rule here stated is quite general and will be equally true if; instead of the one-half, the distance be diminished to the mth part or be diminished by the mth part, pro- vided that the angular velocity at the greater distance be less than required by the conditions of dynamical equilibrium and the diminution in the distance be pro- portional to the excess of active force at the greater distance.. Thus, if at the greater distance the angular velocity be so proportional to the active force (i.e., to the intensity of attractive force) that, supposing the angular velocity to be increased Vw. times, or to be in- creased by the '\/mth part of the previous velocity, the whole of tho force would suffice in ei 'ler case respec- tively to e^uilibriate the centrifugal force, then the ^' A PLANET APPROACHING THE SUN. 51 •distance by the action of the excess of the nttractive force will be diminished to the mth part or be dimin- ished by the mth part (as the case may be), and dynami- cal equilibrium will be attained at that distance, the angular velocity being proportional and the area swent by the diminished radius vector being equal to that swept by the undiminished radius vector at the greater distance. Although the general cnse is necessarily of some com- plexity, nevertheless, the primary facts upon which tlie certainty of the result is dependent are readily intelligible and may be simply stated : (1) If the active force be in excess of what is required by the conditions of equilibrium it will be productive of centripetal motion. (2) The centripetal motion will continue so long as the force remains in excess. (3) Therefore the approach, or diminution of the dis- tance, must be proportional to the excess in the active force by which it is occasioned. (4) That the areas swept by the rrJius vector remain equal although the radius vector be increased (or dimin- ished) is dependent upon and necessarily follows from the fact that the excess in gravitative force, which occasions and determines the amount of the approach, and the diminution of centrifugal force, arising there- from, taken together, suffice to equilibriate an increase (or decrease) in angular velocity exactly equal to the increase (or decrease) in angular velocity which would be occasioned merely by the diminished (or augmented) length of the radius vector supposing the linear velocity to be unaugmented. :ll $$ A PLANET APPROAOilINO THE SUN. (.'5) Thut the conditions of equilibrium are as stated, is u fundamental fact, .cue of the facts of creation .. the law of gravitation and of centrifugal force having been so proportioned that the augmentation of the on«; from the diminution in the distance, is directly proportional to the increase in the other from the accelerated velocity. The fact is demonstrable, by mechanical experiment as to the variation in the centrifugal force ; by astrono- mical observation as to the variation in the gravitative force ; by geometrical observation as to the relation of the orbital area to the angular velocity and length of tlie radius vector. The question, however, to wliich we yet have to reply, does not suppose an excess of attractive fore*; at the outset, but that some extraneous force or impulse occasions the diminution of the planet's distance from the sun to the one-half, and requires to know what will then be the angular velocity of tlie planetary body and whether the conditions of equilibrium wliich existed at the greater distance will still continue to exist. The question is now readily answered, because whatever tlie nature of the extraneous force or impulse may be, its effect, which is to diminish the distance, must in so doing be equivalent to such an excess of gravitative force as would cause an equal diminution ; therefore, since the question supposes the orbital distance of the planet to be reduced to the one-half, the same increase in angular velocity will take place as in the previous case, namely, a fouc-fold increase. But the conditions of dynamical equilibrium will no longer subsist, for, there was no excess of force in the first instance, and, whereas the angular velocity increased four-fold would require IG times the attractive force, tht sctual increase m A PLANET APPaOACniNQ THE SUN. 58 in the attractive force is only 8 times ; consequently the centripetal force will be opposed by double the amount of centrifugal force, and the body will immediately com- mence and continue to recede from the sun until its former orbit at the greater distance is aguui reached. If, assuming the planet's orbital revolution to be as before in dynamical equilibrium, we suppose the dis- tance from the sun to be doubled, tlie previous condi- tions will be reversed, namely, tlie angular velocity will be reduced to the one-fourth, because the mere duplica- tion of the distance, by doubling the magnitude of the circle, will reduce the angular velocity to the one-half without alteration in the linear velocity, and the abso- lute linear velocil y will l^e diminished by the resistance of the sun's attractive force and consequent retardation, which will again reduce the angular velocity to the one- lialf. Now one-fourth the angular velocity requires only the one-sixteentli of gravitative force to equilibriate, and by tlie law of gravitation the duplication of the distance reduces the attractive force, to the one-fourth, but this again is practically reduced to the one-half by the in- crease of the centrifugal force which is proportional to the orbital distance of the body from the centre of revo- lution. Therefore one-eighth the attractive force is opposed by only one-sixteenth the centrifugal force, or, as it will be more strictly correct to express the relation, one-fourth the attractive force is opposed by only one- eighth tbe centrifugal force, and consequently the planet- ary body must ajiproach the sun and return to its former orbit; t.c, tothe lialf distance, where equilibrium will be Jittained. ( 54 A PLANET APPROACIIINO THE SUJf. r>ut npaiii, if wo piipposc, tlie distmice being the same* lis before, tlie planet's orbital velocity to be increased by some sudden cause, in such wise that the attractive force of gravitation at that distance becomes less by the one-half thiin sufficient to e(|uilibriate, (let us take for example the earth, and suppose its velocity increase* (\)li' -V \ I ^ .M" 1/ il THEORY OF THB TIDES. «r force can equilibriate, the effect will be that the orbit will gradually expiHid and the planet will recede in helical curves, the radial distance of the orbit continually increas- ing us shown in fig. 10, until equilibrium is attained. 1.) Elliiticity of the Planetary Orbit. — In the fore-going we have not particularly includi 1 that actual approximation and recession during each revolution of the planet which constitutes the ellipticity of the orbit. The motion toward and recession from the sun may be compared to the oscillations of a pendulum, the alliance between these manifestations of gravitative force being in fact very close. The amount of these oscillations, causing the greater or lesser deviation from the circle known as the eccentricity of the orbit, is, however, partly dependent upon the disturbance (or effect caused) by the gravitating influence of other bodies. But, unless the outside influence or perturbing cause be supposed to act continuously, it is evident that any deviation in th3 one direction wi. be necessarily followed by a compensating equivalent niovement in the opposite, and the average distance of the planet from the sun will remain the same. Hence we may appreciate the stability and pennanent nature of the arrangements whereby each planet is secured from changes in its velocity of motion and relative position, and by whicli the regularity and uniformity of its perioliqiie to the sun's equator. Hence, although Oie orbit is an cUipsi-, the horizontal soction or plane of that orbit miy be m true circle ; and the oscillatiuii, if tliat liu tiiu case, is contineii to the vcr- tKal oscillation of the planet manifested iu its ascent and desoeot abovo Mai. below the horizontal plane. (12.) TU'. INFLUENCE OF THE MOON's GRAVITATION UPON THE EARTH, AND THE THEORY OP THE TIDES. The i.llowing brief and distinct statement of the pre- sently accepted teaching on the subject of the Moon's influence in causing the double tide is from Drew's Mauital of Astronomy. Page 78. " In Fig. 45, let Z. represent the moon, R the earth. Now the moon attracts every particlo of the earth, and, the water, being free to move, will tend towards her at 0.; 4 ■-■: .i J' ^i 1« o it will be -high tide, therefore, to those places situated at 0. and its neighbourhood, which have the moon in the meri- dian ; but since the quantity of water remains the same, the places at N. and S., 90° distant from 0., will supply the liso at p.; with them, therefore, and down the line N.R.S. it will be low water. As the earth turns round with her diurnal motion, other places will advance towards the moon, or will have her in the meridian ; it will therefore be high tide to them at that time. So far iho matter is clear; but tho peculiarity is, that when it is high tide at 0. it will be also high at G. — diametrically opposite, or with those places on whose inferior meridian the moon is situated. To render our explanation of this fact more lucid, let us investigate the operation of attraction on three bodies, at diflFerent distances from the attractive body (Fig. 12). Tho effect of a body y. operating on three others r., z., x., in the samor n: THEORY OF THE TIDES. 69 line, would be to increase their mutual distances ; for r. would be drawn to w., through the space r.w. ; z. being further ott' from y., would bo di'awn through a less space, in the pro- portion yr* . yz*, viz., to v, zv being loss than rw; x would be still less operated upon and would pass through a less space towards the attracting body, viz., xt. The result will be, that the distance of the bodies r. and x. from z. will be increased, r.w. and I'.t., their new distances, being gi'eater than z.r. and z.x., their original distances. Lot the waters on either side of the earth R., in fig. 45, be con- sidered in the same circumstances as the two bodies x. and r. with respect to z. in fig. 12. The operation of the attrac- tion of the moon z. upon them and the earth will be to raise the waters at p., and to draw the earth, as it wore, away from the waters at /•., causing a simultanocus rising of the tides at o. and g." o i a? e — e30 7 \y \^ ir The explanation here given is that the moon attract- ing the water on the earth's surface, at the side next to her, draws it from the earth towards her, and, in addition, attracting the earth itself, draws that away also from the water on the opposite side, thus accounting for the high tide at both sides ; but if such were the case the earth and moon would soon come together ; because, if the larger body, the. earth, deviates froni her orbit* to- • The theory suppobcs the earth to be actually drawn away from the- water ; eridently, therefore, the earth i3 supposed by the theory to deriate- from its orbit and to approach the moon. 60 THEORY OF TUE TIDF.8. i ! approach the moon,., the lesser body, the moon, must deviate so much the more (in inverse proportion to the mass) to approach the earth ; and, as this approximation is supposed to be continuous, the result would neces- sarily be, within a certain limited time, contact between the earth and the moon. Moreover the sun's attraction is supposed to operate in the same manner. Drew's Astronomy, page 79: — "Not only is the moon an agent in producing tides, but the sun also; in consequence, however, of his greater distance his attraction is not so much felt; the whole force of attraction being in compound proportion of the mass directly and the distance squared inversely. The force of attraction thus deduced will give the sun's attraction : the moon's : : 2 : 5." Let us compare with this Lardner's explanation of the phenomena. " Lardner's Astronomy. The Tides. 2514. Erroneous notions of the lunar influence,'^ " There are few subjects in physical science about which more erroneous notions prevail among those who are but a little informed. A common idea is that the attraction of the moon d'aws the waters of the earth toward that side of the globe on which it happens to bo placed, and that, consequently, they are heaped upon that side, so that the oceans and seas acquire there a greater depth than elsewhere, and that high waters will thus take place under, or nearly under, the moon. But this docs not correspond with the fact. High water is not produced merely under the moon, but is equally produced upon those parts most removed from the moon. Suppose a meridian of the earth so selected, that if it were continued beyond the earth, its plane would pass through the moon, wo find that, subject to cer- .lain modifications, a great tidal wave, or what is called XnEOBY OF THE TIDES. 61 "high water, will be formed on both sides of this meridian ; that is to say, on the side next the moon, and on the side re- mote from the moon. As the moon moves, those two great tidal waves follow her. They are, of course, separated from each other by half the circumference of the globe. As the globe revolves with its diurnal motion upon its axis, every part of its surface passes successively under these tidal waves; and at all such parts as they pass under them thero is the phenomenon of high water. Hence it is that in all places there are two tides daily, having an interval of about 12 hours between them. Now, if thecommon notion of the cause of the tides were well founded, thero would be only one tide daily, viz., that which would take place when tho moon is at or near tho meridian." 2515. 7*716 moon's attraction alone will not explain the tides. — That the moon's attraction upon the earth simply consid- ered would not explain the tides is easily shown. Let us suppose that the whole mass of matter on the earth, includ- ing the waters which partially cover it, were attracted equally by tho moon, they would then be equally drawn towaixli. that body, and no reason would exist why they should bo heaped up under the moon ; for if they were drawn with the same force as that with which the solid globe of the earth under them is drawn, there would be no reason for supposing that tlio waters would have a greater tendency to collect toward tho moon than the solid bottom of the ocean on which they rest. In short, the whole mass of the earth, solid and fluid, being drawn with the same force, would equally tend towanl the moon ; and its parts, whether solid or fluid, would preserve among themselves tho same relative position as if they wore not attracted at all. 2516. Tides atused hy the difference of the attractions in different parts of the earth. — When wo observe, however, in a mass composed of various particles of matter, that the relative arrangement of these particles is disturbed, some" 1^^ i m 62 THEOHY OP THE TIDES. " being driven in certain directions more tlian others, tiie infcronco is, that tlie component parts of such a mass 1191st be placed under the operation of different forces; those which tend more than otliers in a certain direction, being driven with a proportionally greator force. Such is the case with the earth placed under the attraction of the moon. And this is, in fact, what must hajipen under the operation •of an attractive force liUe that of gravitation, which dimin- ishes in its intensitj- as the square of the distance increases. Lot^., B., a, D., E., F., G., H, fig. 731, represent the globe of the earth, and, to simplify the explanation, lot us first suppose the entire surfji^e of the globe to bo covered with water. Lot M., the moon, bo i>laced at the distance M.H. from the nearest point of the surface of the earth. Now it .If. •will bo apparent that tlie various points of the earth's sur. face are at different distances from the moon M, ; A. and G- are more remote than 11. , B. and F. are still more remote ; C. and E. m.jre distant again, and D. more remote than all. The attraction which the moon exerci.ses at //. is, therefore, greater than that which it exercises at A. and Cr., and still greater than that which it produces at B. and F.; and the attraction which it exercises at D. is least of all. Now this attraction equally affects matter in every state and condition. It affects the particles of fluid as well as solid matter, but there is tliis difference, that where it acts upon solid matter the component parts of which are at different distances from it, and therefore subject to different attractions, it will not disturb the relative arrangement, since such disturbances or disarrangements are prevented by the cohesion which " THEORY OF TirE TrDES. 63 "chnrnctorizof* a solid body; but this is not the case with fluidH, the particles of which are mobilo. The attraction which the moon exercises upon the shell of water, which is collected immediately under it next the point Z., is greater than that which it exercises upon the 8olid mass of the globe ; consequently, there will be a greater tendency of this attraction to draw the fluid which rests upon the surface at //. toward the moon, than to draw the Kolid mass of the earth which is more distant. As the fluid, by its nature, is free to obey this excess of attraction, it will necessarily heap itself up in a pile or wave, over //., forming a convex protuberance, as repre. ficnted between r. and i. Thus high water will take place at 7/., immediately under the moon. The water which t'<>is collects at H. will neces arily flow from the regions B. and F., where therefore there will be a diminished quantity in the same proportion. But lot us now consider what happens to that part of the earth D. Hero the waters, being more remote from the moon than the solid mass of the earth under them, will be less attracted, and consequently will have a less tendency to gravitate toward the moon. The solid mass of the earth D.n. will, as it wore, recede from tho waters at n., in virtue of tho excess of attraction, leaving tliese waters behind it, which will thus bo heaped up at n., so as to form a convex protuberance between c. and A., similar exactly to that ■which wo have already described between r. and i. As the difibro'^co between tho attraction of tho moon on tho waters at z. and tho solid earth under tho waters is nearly the f»amo as tho diffbronco between its attraction on tl»o latter and upon tho waters at n., it follows that tho height of the fluid protuberances at 2. and n. are equal. In other words, the heights of tho tides on opposite sides of the earth, tho one being under tho moon and the other most remoto from it, are equal. J THEORY OP Till TIDES. " It appenrn, therefore, tlint the cause of the tides, ao far ns the action of the moon in concerned, in not, oh i» vulgarly HuppoBod, the mere atLruction of iie moon ; Hinco, if that. attriK lion were cfjual on all the component parts of the earth, there would asHuredly he no tideH, Wo are to look for the t-autse, not in the attraction of the moon, hut in the iinquutiiji of itri attraction on different parts of the earth. The greater this ineriuality w, the greater will he the tides. Hence, uh the moon iu wuhject to a Hliglit variation of dis- tance from the ear''), it will follow, that when it la at its least distance, oral lie point calh-d perlijce, the tides will ho the greatest ; and when it is at the greatest distance, or at the jM»int calleecome,s ai»i»arent that in n-Hjiect to the Hnn's attraction thewlmle earth, water and hind, is in dynamical e(|Milihrinm ; that is to say, the centrifugal force of the whole earth and of each and every j>art tliereof, solid and fluid, counter- halances the attra«;tivc fore*; to which it is subjected. There is consecjuently no reasonable ground shown for attributing the secondary (so-called solar) tide to the attraction of the sun.* A probable explanation may be found in the 8U[^ sitiofi of u secondary tide arising from • We do not Pay thnt tlic siiii'h in/hioiice may not eonietiinoH |>ri)- diice an npprrciulile tidal eticct hy inleiiHitying or diniiniHlting tlie indnence of tlic moon, Ijiit tlie Sjiring and Nea|) Tides may Ix; attri- liiited with iiioreprolmliility toanotlicr cause. When it ifl underHtixxl that the earth iiHceiids and descendfi thronjih a vertical space of aljoul 47 decrees during! each annual revolution, and that the velocity of this vertical motion is greatest at the time when tlie earth passes through, or is near to, the plune of the sun's equator, it will Ijecome u])parent that (he dynamical condition of the waters on the earth's .•surface, hence arising, being interfered with hy the action of the moon may account for the ed'ect of the moon's influence being augmented or diminished about the time of the equinox. When we come to the particular consideration of thin case, which we call • the theory of the earth's jierpendicular axis,' it will be seen that the theory applies ^-nerally to all memlters of the solar system, satellites as well aa planets. Hence, the moon having also a vertical motion of ascent and descent above and beneath the plane of the earth's equator, the local eflects of her influence on the earth's surface must be modified TIIKORV or TUB TIDRH. C7 tilt' y the imooii'h iiiflii- ciicc ; tilt; water tliiiH n('(|iiiriii/^ a inovtMiicur of OHcilla- ♦ ioii wlii<;li n'HiiltH in a ^'ri-at vvav« Kiipplnin'iitary to that (l by (U'litrifiigiil force, may he rca8onal)ly 8ii|([(os»'(l to iiillucncu tlic \vat«'rs on the earth's 8urfac«', l»y her attra<'-tiv« force causing them to ac(;umu1ate on that side of the earth next toiler, and thus, by increasing the depth, occasion the great tidul wave as explained by Ldrdticr and Drew. Hut the explanation in respect to the Kimultaiieous wave on the opposite side of the earth must b»! objected to as (piite unreasonable, for, as already stated, it includes the as- suiii[>tion that the earth is actually moving nearer to the moon, and therefore that the earth and the moon are continuously approaching each other. The cnse may be thus exidained. Let us consider what the etlect would be siqiposing a local increase of gravitative attraction to take place beneath a part of the earth's surface. For instance, let us suppose such increase confined to 2,000 square miles, and that the augmentation of the attractive force is considerably greater in the central part of this space and diminishes towards the boundary where it becomes equal to that exerted on all other parts of the earth's surface. Now •:'' 1 I 'i !' !"' tlieri'liy. Wlicii llie variation in tlie comlitions causeil by tliiH verti- . te .* h ?i '* •Owes I - 8 i, - f 5- s- s O B 3 a' " E? s" J ^ « I" « Mir n. 3 « • CD <«• Bs. •» -. P o c; s. "•• -" S P I c >■ -( e. o a m N S3 § ^». 2 H <-*. a 1" w A o "^ w the Mooti surface ^ o H ^ c. a Si^ w «. a o ^ S. d 5^* 2 w r K o • 8 f ' 1 f ' 70 THEORV OP xriE TIDEH. I !l Fig. II. — Tiie (lispliM't'iiit'iit of the eiirth's centre of gravitation (Voin (■. to X in the rliulanciii|>; the cnrtliV ^rnvitation ; tliiiK, tiko coluiiiii of the water Riilijictcil tu the liiiinr iiitliu>nui> is ren. (lori'il lighter, ami forccil ii|iwiinlrt hy the heavier water not ho ititlii- enoeil;.. and, oet ihe vippofite Hiile, the parlieieH of water CMWil together where th' ultraetinn is stroiige.-t until the surface attraelioit iH ef|uallzc miles. Anil aince tho attrac jW of gravity docreasou in tho in- verse pro|H>rtion of tlie Ht^iiaro of tlio diNtanco t'roni tiiu evntro of tiio attraeting Npiiero, il is phiin thai tlio attrac- tion will be weaker on tiio oarth'ii surtm-o near the eiiuator than at or near the poles in the proportion of the hi[uaie of half the earth's cfpiutorialdiainiter to the sipiaie of half the polar diameter." Tills statiMiicnt would have nmcli lorcc if the eurth's soiii'cu of gravitation was actiniliy located at and acted from a point or tmiall splu^rical place Hituatd at the earth's centre. For nniny [inrposes it is cunveiiient and does not practically involve any error to refer the njrgre- gate or collective uctioti of gravitation to the centre ; since, if vve assntne the eurt'i to be a perfect sphere, the attractive force wonld be, everywhere on or beyond the snrfuce, eond o.v...'flv witli it. This al)sonce of com- plete corrcspon;r«''.s' Mimntil of Asfrovom;/: "The ])endulum offers to us the means of determining this point with great accuracy; for bv marking the number of vibrations in a giv>>n time at dilferent |)ointson the surface ot' the globe, the force of gravity nui}- be de- tected ; it will, in fact, be as tho square of the velocity of tho vil»ration« in the Kovcnd place> ; or it may be deduced from Iho length of tho bvcoiidti pendulum at various distances m 'it THE PENDULUM AS A MEASURE OP (jHAVITV. 75 " from tho o |uator. OlwervutioiiH to this oflect linvo hoon mado in various latitudes, and the result aj^rees witli theory, in nssi^'nin^ llie ,,';, as tlie loss of gravity at tho of|uator com- pared with what it would Ito at tiie poles; so that a body weighing 194 Ihs. in the latter |iosilion would only weigh 193 at tho cqiwitor ; of this quantity arising from the two causes nlM)vc assigned, j^'^ part must he attributed to tho s]))ioroi- dal (iguro of tlic earth, ^j^ to tho centrifugal furco, and sso ^ ^fii< "** fthout I-}, J to both causes combined. Tho pntpovtion lietween tho earth's polar and c(|uatorial i Miiw iiiii mwii Mirw * '* f« THE PENDULUM AS A MEA8I RE OP atUVixy. it appears that in respect to the application here made an inference or assumption is included which is certainly not justified or supported by fact The assumption to which we olytfct is thus stated by Dr. Larduer in his Handbook of Natural Philosophy: " 540. Time of oscilhtion varies with the attraction of fjravity. — Since the f'orco which producos the oscilhition of a pondulum Ih Uio accelorating forte of gravity urging the pendulous body alternately from the oxtromities of the arc of oscillation to the middle ])()iiit of that arc, it is evident that if this force were increased in its intensity, llie velocity with which the pendulous body would lie precijiitated to its lowest position would bo increased, and conHe<|uently tho time of oscillation diminished; and if, on the other hand, tho inii)olling force of gravity wore diminished, the force urging the pendulous hotly being enfeebled, it would bo moved with a diminiHhed velocity, and, consequently, tho time of OHcil- lation woulil bo increased. It follows, therefore, that tho same pendulum will oscillate more slowly or rapidly, aecoi-d- ing UM tho force of gravity which acts upon it is diminished or increased. '•541. Law 0/ this variation. — But it is not enough to state that a variation in the force of gravity will change the time of o.scillati(in of the pondnium. It is required to ascertain in what proportion it will produce this change; that is to say, if the force of gravity acting on tho pendulum be augmented in any given ratio, in what corresponding ratio will tho time of oscillation of such pondulum be diminished. It is proved in the theory of accelerating forces, that under such circumstances, tho squares of tho times of oscillation will vary in tho inverse proportion of the force; that is to say, in whatever ratio tho force of gravity bo augmented, tho squares of the times of oscillation of tho pendulum will be diminished in tho same ratio.' I *fl TEHRRNTHIAL ORAVITATION. 77 The nssumption is cxproHMfd in tlin last [»)iragru|;!i, iiuiuely, ' tliatthc si|uari>8 of tint Iiiiicn of oMcillution will vary in tlie inverse piopoitioa (>( the (oive.' Now thJN is not in acconhnice with tl^' law of ufoehTutcd motion uiidur thu ittihit'iKHMtl a loive Niieh as thni «»( ^lauta• tiun. . the hiw is (hat the ttiiie oceupiei) by the l»o«ly in niovim^ {i.e., {'i\\\\\\i^) tlwouirh a certain H|»ac(,' i« iiverselv |Moi>ortional U\ \\\\^ loree simply. . so that if tlie force of gravity he augmented, die time, and not ih., 299 : 29f-i, for if the eHect were not les- sened by the increased distance from the earth's '-entre, the augmentation of the forie wouhl be as the difference in the cubes of the proportional lengths of the dia'm'»«'rs ; but as, by the law of gravitation, the force i» diminisiied as the square of the increased distance, the result is iin increased effect equal to the increase in the diameter sinj- I' I T8 FORCE OK Till ORAVITATINQ MArtH. ply; consecmeiitly, at the equator, we hiive tliis augmen- tation in the gravitative force to Met otl" against the dimi- nution eaused hy the counteracting effect of the centri- Ciigal force. Wherefore, taking the centrifugal force as given hy Drew at ,^5, and the increase of gravitative force from the su|>erior diametrical length at gi,, we have av. - aii=4al,-„i »H tlie theoretical quantity, thus rougldy estimated, to compare witli the (corrected) result of the pendulum exp«'riments ; a very sliglit discrepancy considering tin* mnnljer of slight interfering causes which might (M)ssihly affect tlie appan^it result, on the one liand, and tiie precision reipiired in all tlie data to obtain the result with perfect accuracy on the other. (lo.) TlIK UESLLTAXT ATTKACTIVE FOHCK OK THE COLLEC- TIVE GRAVITATINO MASS. It is convenient and, for some purposes, not incorrect to refer the forces emanating from all the parts of u spherical mass of matter to a central point, but it should be remembered that in fact the force belongs to the s]>here generally, and emanates from every part and point in the sphere, and the force acting on the body at its surface is exerted not only in the vertical direction, but also in each and every angular direction in which a line can be drawn from any part or place in the sphere to the body on the surface. The body on the surface is not, however, acted upon with an ecjual amount of attractive force from every part or jioint in the sphere,- but the amount exerted by ports of efpial mass is dependent upon the situatitn of the part relatively to tlie centre of the sphere and to the body on the surface ; the actual pro- portionate amount of influence is measured by the angle contained between a line connecting the body with the COMPOSITION OF TERRESTRIAL ORAVITATION, 7a Cf'iitro, ninl a lino connoctiiig the body with the part whoiwe thr iiifliUMice h exerted. A little foiisideratiou will show tliiit only tlione |iiii-t8 dirntly in the line from the luidy pjissiiii; throujxh the centre of the sphere can exert tlieir entire infhu'iue on the l«ody, that is, the whole of the direet inflnenee hehtnifing to their distanee Iroin the body ; from all other parts of the Hphere the inllnence [jroceedinj; from any part on one side of the central line, is in a irreateror lesser nieasiire opposed hy an e(|nal inllnence proceedini; from the similarly sitnated part on the other side of the line; and t.iis opposition will become greater and the actinix inflnenee less, directly in proportion as the anule, contained betwe.>n the line connecling the body on the surface with the centre and the line connecting the body witli that point whence the inflnenee proceeds, becomes greater. This is illnstrated at fig. 11, where/, represents a body on the surface of the "phere ; /. e. the line comiecting the body with the centre; /./., f.h., /.k., f.f/., lines of force through which the body /. is acted on by gravitation from the points /., /<., h.j (J., respectively. Any part (every part) in the line///, is opposed by a similar part in the line /.A. ; of these m.f. and n.f, in the side figures represent that proportion of the influence exerted, which is directly opposed and neutralized from the opposite side, each by the other; and f.h. and /.{'. that proportion of each which is directly eflective on the body at/, the same as if that portion (of the force represented by the line /p., or the line //*.) i>roceeded from a point situated on the central line. .Similarly in the upper side figures, k.f. and «./., representing the influence of parts and points situated in the lines k.f. and //, of the centre figure, are I IMAGE EVALUATION TEST TARGET (MT-3) /. 1.0 ■^ iU i2.2 J!f Ki i I.I 2.0 1.8 ||l.25 |,|,.4 m < 6" - ► HiotQgrafiiic Sciences CbrporatioR \ iv> "^ % ;\*^ \ 23 WIST MAIN STMIT WiBSTIR,N.Y. 145M (71«)t72-4S03 i\ m 60 COMPOSITION OF TERRESTRIAL GRAVITATION. resolved each of them into two forces, active ot right angles to each other, of which the one part directly opposes the similar part of the force acting on the body at the same angle from the opposite side of the central line (as m.f, and n.f.), and the other part acts directly in the vertical line towards the earth's centre as /J. and/e. m It The case may be thus illustrated: — Supposing two planets, each of them the size of the earth, were brought sufficiently near to each other, they would both move with equal velocity towards each other until they came into contact; and if, instead of equal sixes, one of them was half the size of the other, then the lesser would move towards the greater with twice the velocity with which the greater approached the lesser. Now, if we take two spheres of iron weigliing (say) ten tons each, and place them a short distance apart ; will they (rush together) move towards each other with equal velocities ? Or, if the one weighs twenty tons and the other ten tons, will the lesser approach the greater ? No, — in either case the two bodies will remain apart; and no tendency even, to fall towards (or approach) each other will be apparent. Let the size of the larger be very gi'eatly increased and the size of the smaller decreased, do we then find any apparent tendency in the lesser to approach (fall towards) COMPOSITION OF TERRESTRIAL GRAVITATION. 81 the greater ; for example, let us take a mountain of con- siderable height and having one of its sides nearly perpendicular, and near to that side and from the same height (as that of the mountain) allow a piece of iron (or other such body) to fall to the ground ; — will the piece of iron in falling approach in any degree the mountain ? It is well known that it will approach the mountain. Let us then again carefully consider the case of the two iron spheres under the actual conditions and circumstances belonging to it. Fig. 12 represents the earth. In order to imagine the two spheres visible we will increase their mass and take the larger (a) at 2000 tons,andthe lesser (6) at 1000 tons.* By drawing a few of the lines indicating the direc- tion in which various parts of the mass compounding the earth exert their gravitating influence, it at once becomes apparent that the attractive force of the larger sphere is opposed from the directions/, g, h, i, &c., and is aided from the directions m, n, o, p, &c. ; it is true that these oppos- ing forces from the earth's mass would of themselves neutralize each other and leave a certain amount of vertical attraction (apparently from the centre) as the lesultant, but when an additional (outside) source or subject of gravitative influence becomes effective, the lii • NoTB— In fact on the scale of the figure a sphere of 2000 tons would be a point so small as to be scarcely visible. (This has reference to a figure of 18 inches diameter.) 83 COMPOSITION OF TEKRESTRIAL GRAVITATION. My v/hole must be' considered together and seperately, i.e,, the whole as compounded of its parts, in order to appre- ciate the actual effect upon the lesser sphere &, and it becomes evident that the resultant will be a very slight deviation in the Hne of the apparent attraction of the earth, a deviation which (by , enormously exaggerating the effect for the sake of illustration) may be indicated by the line Z*, 'c, forming the angle 'c, 6, c, with the ver- tical central line of attraction. When it is considered that on the scale here shown tlie size of (a) would repre- sent a number of the largest mountains on the earth's surface united into one sphere, it will be readily under- stood that if (a) be reduced to the size of any ordinarily very large object the effect which its gravitative influence could have upon another oL ct of the same or less size than itself would be almost or quite inappreciable, and, notwithstanding this, the actual gravitative influence exerted by it may be proportionally just as gi-eat or even greater than that of tlie earth; and, if the earth's influ- ence were removed, these two spheres would approach each other (rush together) with a velocity respectively proportional to their bulks ; if of equal size they would approach each other from a proportionally small distance with the same velocity as that with which the two planets of equal size would approach ; if the one was twice the size of the other, then their respective velo- cities would be as two to one.* ii| * Note. — Proportional distance ;— if the planets were of the earth's size, half a diameter would be 4000 miles; if the two spheres were one foot in diameter, half a diameter would be 6 inches. 1 (10.) GRAVITATION AND THE ATOMIC THEORY. The expressions ' densities,' ' specific gravirtes/ 'atomic weights,' may be used without a clear per- ception of the distinctions and relations between them. If a body in the solid condition, capable of undergoing compression, be subjected to a sutHcient pressure, its bulk or volume will be diminished and a proportionate increase take place in its density ; its weight or gravity, therefore, relatively to its hulk will have become greater. Hence, specific gravity has a direct and very close rela- tion to density, and, for any one body relatively to different conditions of itself, the one term is so entirely dependent upon the other that the terms, in such limited relation, may be considered almost synonymous ; in fact, in such a case, the specific gravity measures the density, and vice versa : but if the comparison be made with another body, composed of a different kind or variety of matter, the same necessary inter-dependence no longer holds good ; because the equal bulk of the second body may have (and the variety of m.itter being different, it will almost certainly have) a diflferent atomic weight; and, consequently, although the first body may be in its most dense, and the second in its least dense con- dition, the specific gravity of the second may be, never- theless, greater than that of the first. The intensity of the gravitating influence at the surface of the earth may- be measured by the velocity acquired or the space passed through in a definit.-? time by a falling body. The law which governs and regulates the motion and progress of a body so falling belongs to the law of gravitation ; and the conditions which accelerate or retard the descent of a falling body have been investigated with considerable; attention and care. ill ■f,i «M u SPECIFIC GRAVITY. One of the facts coiKjlusively ascertained is, as pre- viously stated, that neither an increase in the density of a body through contraction, nor decrease through expansion of the volume, nor yet an addition to the mass (that is, an addition to the quantity of matter con- tained in the body) makes any difference as to rapidity in the descent of the falling body ; the motion is neither accelerated nor retarded : the velocity is the same. But : — What if there be a difference in the atomic weights between two bodies I Will that make no difference in the relative velocity of their descent ? Supposing there is a considerable difference between the atomic weights of two solid bodies, and that both of them are allowed to fall from the same definite height to the ground : — Will they reach the ground in precisely the same time ? A well known experiment having this question for its especial subject, is recorded by Dr. Lardner as follows : — Lardner's Natural Philosophy, page 108. (236) " Guinea and feather experiment. — Let a glass tube A.B. of five or six feet in length, be closed at one end B. and supplied with an air-tight cap and stop-cock at the other end A. The cap being unscrewed, let small pieces of metal, cork, paper, and feathers be put into it, the caps screwed on, and the stop-cock closed. Let the tube be rapidly inverted, so as to let the objects included fall from end to end of the tube. It will be found that the heavier objects, such as the metal, will fall with greater, and the lighter speed, as might be . expected, this difference of velocity in , e, not to any difference in with less But that falling is the opera- B KJ SPECIFIC GRAVITY. 65 " tion of gravity, but to the resistance of tlio air, is proved in the following manner. Let the stop-cock bo screwed upon the plate of an air pump, the cock being open, and let the tube be exhausted. Let the cnck then be closed, and un- screwed from the plate. On rapidly inverting the tube, it will bo found that the feathers will be precipitated from end to end as rapidly as the metal, and that in short, all the objects will fall together with a common velocity." Page 109. " Weight of bodies prop'irtional to their quanti- ties of matter. — Since the attraction of the earth acts equally on all the component parts of bodies, and since the aggro- gate forces produced by such attraction constitute what is called the weight of the body, it is clear that the weights of bodies must be in the exact proportion of the number of particles composing them, or of their quantity of matter. Hence, in the common affairs of commerce, the quantities of bodies are estimated by their weights. It will appear, hereafter, that the weight of a body, or the force with which it is attracted to the surface, is slightly diiforent in different places upon the earth ; but this is a point which need not bo insisted upon at present. At the same place the weights are invariably and exactly proportional to the quantities of matter composing the bodies. If one body have double or triple the weight of another, it will have double or triple the quantity of matter in the other." Since a case involving consideration of the elementary particles of compounded or aggregated matter belongs also, and especially, to the domain of Chemical Science, let us refer to a reliable exposition of the teaching on the subject now accepted by the chemist. Fowne^s Manual of Chemistry, page 189. (3) Law of Equivalents. — " It is highly important that the subject now to be discussed should be completely understood. Let a subject be chosen whose affinity and powers of com- bination are very great, and whose compounds are suscep- " 86 THE ATOMIC THEORY. tiblo of rif,'id and exact analyisis ; such a body is found in oxygon, which is Icnown to unito with all the olomontary substances, with the single exception of fluorine. Now, let series of exact experiments be made to determine the i)ro- portions in which the different elements combine with one and the same quantity of oxygon, wliich for reasons he-o- after to be explained, may be assumed to be 8 parts by weight ; and lot those numbers be arranged in a columu opposite the names of the substances. The result is a table or list like the following, but of course more extensive when com- plete : Oxygen ,.,. 8 Hydrogen 1 Nitrogen 14 Carbon 6 Sulphur 16 Phosphorus 32 Chlorine 35.5 Iodine 127 Potassium 39 Iron 28 Copper 31.7 Lead 103.7 Silver 108 &c., &c. Now the law in question is to this effect : — If such num- bers represent the proportions in which the different ele- ments combine with the arbitrarily fixed quantity of the starting substance, the oxygen ; they also represent the pro- portions in which they unite among themselves, or any rate bear some exceedingly simple ratio to these proportions." Page 193. Combination by ijhime. " The ultimate reason of the law .a question (combination by volume) is to be found in the very remarkable relation established by the hand of Nature* between the specific ♦ The talented author, Prof. Fownes, of whose writings one is an essay on ' Chemistry as ezempiifying the wisdom and beneficence of God, means by "the band of Nature "...the Hand of the Maker of Nature {Kuklot.) CHEMISTRY AND PHYSICAL ^CIENCE. 8T * gravity of a body in the gaseous state and its chemical •equivalent; a relation of such a kind tliat quantities by weight of the various gases expressed by their equivalents, or in other words, quantities by weight which combine, occupy under similar circumstances cf pressure and tempe- rature either equal volumes, or volumes bearing a simple proportion to each other. If both the specific gravity and the chemical equivalent of a gas be known, its equivalent or combining volume can bo easily determined, since it will be represented by the number of times the weight of an unit of volume (the specific gravity) is contained in the weight of one chemical equivalent of the substance. In other words, the equivalent volume is found by dividing the chemical equivalent by the specific gravity." If we consider the elementary atoms of the chemist to be the elementary particles of matter, then, it is quite evident, that these results, of very numerous carefully conducted chemical experiments, entirely disagree with the deductions from the guinea and feather experiment previously detailed ; because the information furnished U9 by these experiments is that the weight of a body consists in the atomic weight of its elementary particles multiplied into the number of those particles ; or, in other words, the atomic weight of that particular des- cription of matter of which the body consists multiplied into the quantity thereof. Now we have seen that the physicist, as represented by Dr. Lardner, has unhmited confidence in those general- izations and conclusions which based upon mechanical experiment are considered by him as the exposition of an established law. Does the chemist show equal confidence in the atomic theory, and in those experiments upon the 8$ THE ATOMIC THEORY. jii; results of which its title and claim to confidence are founded ? Fowne's Manual of Chemlxtri/, ptuje 200. — " Tlie theory in queHtion (tho atomic theory) has rendered groat Borvico to chemical science, Ac, &c." " At the same time, it is indis- pensable to draw tho broadest possible line of distinction between this, which is at tho best but a graceful, ingenious, and in its place, useful hypothesis, and those great general laws of chemical action which are the pure and unmixed result of inductive research." * So that the atomic theory is not only considered incon- clusive, but it is thought proper to caution the student to look upon it with a sort of distrust, as being, at best, only a graceful and ingenious hypothesis. To show that we entirely dissent from this teaching on the subject, we will here express our belief that the atomic theory is, and has been for some time past, virtually a demonstrated theorem / and, as such, shown to be a com- 2)0und fact, — the great fundamental fact upon which the structure of chemical science rests. It is true, it has not been as yet, formally demonstrated ; but that is, appa- rently, because no one has taken into consideration the possible consequences direct and indirect of leaving a science such as chemistry without any demonstrated and acknowledged basis. One of the consequences is the caution which it is considered necessary to give to the student, as above. There might be, however, an objec- tion to admitting the atomic theory (and the law of com- bining equivalents) as a demonstrated theorem, side by " Note. — The expression atomic weight is very often substituted for that of equivalent weight, and is, in fact, in almost every case to be- understood as such : it is perhaps better avoided." SPECIFIC GRAVITY. 89 ' side witli the (no called) law of natural philosophy, (pre- viously stated) based on mechunicul experiment ; because if understood in the usual sense, one of these laws evidently, to some extent, contradicts the other. The gu'nea and feather experiment can scarcely be considered of so refined a character as to be conclusive on the question whether the velocities of various kinds are absolutely equal when falling to the ground under the influence of gravitation. We may, however, deduce the result from the results of reliable experiments which have been already tried and recorded. The specific gravities of the various metals have been carefully deter- mined. A cubic inch of (cast) iron weighs 4.17 oz. A cubic inch of (cast) lead weighs G.37 oz. Now, if we take a cubic inch of each of these metals, and, connecting the two weights by a fine line, suspend them in an Att- wood's machine by passing the connecting line over the grooved wheel, — we can say with certainty what will happen, viz., the lead weight which is more than 2 oz. heavier than the other will descend with a considerable and a continually accelerated velocity ; and, in doing so will raise tiie iron weight with an equal velocity. If the two weights are now detached and allowed to fall from the same height to the ground — will they descend with an equal velocity ? No doubt they will ; because in doing so the equally rapid descent of the heavier weight will represent a larger quantity of eflfect exactly proportionate to the preponderance of weight. Does this decide the question ? Dr. Lardner's infer- ence is that the quantity of matter or number of particles contained in the lead is greater than in the iron, and therefore both of them descend with the same velocity j o 1)0 THE ATOMIC TIIKoRY. but, taking the atomic theory, are we thereby taught that an at(Mn of lead is of preciwely t!ie same weiglit as an atom of iron, or of gohl, or of potassium ? Does the cubic inch of lead contain a greater number of element- ary particles of lead than the cubic inch of iron contains of the elementary particles of iron ! It is evident that the deduction of Lardner becomes or includes a primary definition of matter; in other words it includes the corollary . . . that if a. and h. represent two distinct varieties of matter, and the combining equi- valent (or atomic height) of a. is tv^'ice that of h., tlie elenientary atom of a. contains twice the quantity of primary matter (i.e., of matter in a more simple and elementary form) contained by the elementary atom of b. If the elementary atoms are of the same size, then that of a. must have twice the density compared with that offc. • • * The important distinction herein defined is. .that gra- vity is not a property of which one variety of matter pos- sesses more or less than others, but belongs to a primary form or condition of matter ; and that a fundamental difference, between all those varieties of matter known to us, is that the elementary atom of any one variety, is com- pounded of a greater or of a lesser quantity of primary matter than the elementary atom of any one of the other varieties. We are strongly of opinion that such conclusion may be demonstrated and established, and, with such inter- pretation and definition, the law stated by Lardner, and the ' atomic theory ' harmonize perfectly. (17) The Hahitahility of the Moon The possibility of the moon being inhabited by men iind other animals is in the first place dependent on the moon having nn atmosphere, and which, since astronomi- cal observation has made known that on the side next the earth there is no atmosphere, is dependent upon t!ie condition of that side of the moon which is constantly turned away from us. Sir John Ilerschel prefaces some opinions on the sub- ject with the following remarks. . . . Outlines of Astronomy, paye 287. "On the subject of the moon's habitability, the completo absence of nir noticed in art (431), // general over her wholo Huriaco would of courso bo deciHivo. Some considerations of a contrary nature, however, mggest themselves in conse- quence of a remark lately made by Prof. Hansen, viz., tliat the fact of the moon turning always the same face towards the earth is in all probability the result of an elongation of its figure in the direction of a line joining the centres of both the bodies acting conjointly with a non-coincidence of its centre of gravity with its centre of symmetry. To the middle of the length of a stick, loaded with a heavy weight at one end and a light one at the other, attach a string and swing it round. The heavy weight will assume and main- tain a position in the circulation of the joint mass farther from the hand than the lighter. This is not improbably what takes place in the moon. Anticipating to a certain extent what he will find more fully detailed in the next chapter, the reader may consider the moon as retained i^ her orbit about the earth by some coercing power analogous to that which the hand exerts on the compound mass above described through the string. Suppose, then, its globe mado up of materials not homogeneous, aad so dispersed in its in- terior that some considerable preponderance of weight i i tj i !l I: 92 HABITABILITT OP THE MOON. should exist eccentrically situated ; then it will be easily apprehended that the portion of its solid contents, under all the crcumstances of a rotation so adjusted, will permanently occupy the situation most remote from the earth." The experiment (or instance) upon which the fore- going Jjypothesia is based, is, we think, clearly open to objection. The result of such an ex^jeriment would only have a reasonable application to the case of the moon connected by gravitation with the earth, if the string was attached to the centre of gravity of the loaded stick- But if the experiment were to be so tried it is not doubtful that tha result would be opposed to the hypo- thesis of Prof. Hansen; because the body, whether it were the arrangement consisting of the stick and weights, or the moor, would retain the position in which it might happen to be, or in which it was placed, at the time the motion of revolution was given to it ; and, if an impulse of rotation were to be also communicated to it, it would continue to rotate around its own centre of gravity whilst travelling in the orbital path around its primary centre of revolution. In the remarks as to the physical conditions on the moon's surface. Sir John Herschel supposes the existence of both water (in the liquid form) and of an atmosphere ; all the water, and almost all the atmosphere, being con- fined to one hemisphere, in consequence of a superior gravitative force on that side furthest tirom the earth '^ but the form of the (solid) moon is not supposed to deviate very considerably from a sphere, and it is there- fore not apparent how the hypothesis of the existence of the water and air on the one side only is to be recon- ciled witii the circumstarces of the case; for instance^ HABITABILITY OF THE MOON. 95 the average intensity of gravity on the surface of the moon compared with the intensity on that of the earth is about 4 : 15, that is, if the earth's gravity is represent- ed by 15 lbs. on the square inch, that of the moon w ji Id be about 4 lbs. on the square inch. Now the moon's atmosphere is supposed to be proportional in quantity to that of the earth ; it would be, therefore, about one- fourth the depth (or height). Since 15 lbs. on the square inch is the weight of the greater column, acted on by the earth's more intense influence ; what will be the pressure caused by the lesser colu.nn only one-fourth the height, acted on by the (moon's) influence less in- tense in the proportion of 4:15! When it is understood that, on the supposition of a uniform distribution of water, together with the quantity of atmosphere conjectured, the water would be subjected to a pressure only the one-fifleenth of that on the surface of the earth ; and that air at the level surface of the moon would be far less dense than at the summit of the earth's highest mountains ; the very large amount of concentra- tion of gravitating influence which would be required in the one hemisphere to fulfil the conditions belonging to the hypothesis becomes apparent. We may suggest that by supposing a much greater alteration of form, and as- suming that the side of the moon furthest from us may be more or less concave instead of convex, the conditions would be then such that the existence of water in the liquid form and of animal and vegetable life (such as knovn to us) would be possible, and not perhaps (vio- ientl), improbable. The effect Vould be to give such a concentration as the circumstances require. Fig. 13 . . 9i HABITABILITY OF THE MOON. a. and b. may serve to convey a general idea of the ar- rangement supposed ; a. being a vertical section of the moon through the centre (i.e., by a plane passing through the centres of the moon and earth) and (6.) a surface view or plan of the side furthest from us. The rainfall is supposed to take place on the high land forming the side of the concavity (which may be supposed to consist of a circular chain of rocky mountains), whence, collect- ing into rivers and streams and watering the intervening habitable country, it would flow into the central ocean, therein again to undergo evaporation from the influence of the sun and of the internal temperature. A great aerial current ascending in the central, and descending in the higher lateral regions {i.e., the circular boundary) would assist and regulate the effect, and the atmosphere itself would be concentrated and retained (at least in a great measure) within and above the concave side. The central ocean may be supposed to contain large islands as indicated in the figure ; to be entirely open ; or, to be divided by land into several large lakes or seas. The diameter or breadth of the central ocean may be taken at about 800 miles : and the habitable land sun'oundinsr it to have a breadth of about 500 miles on each side j which would give, in round numbers 500,000 square miles of ocean surface, and 2,000,000 square miles habit- able land surface. 2 1 HABITABII.ITY OF THE MOOX. 95 a. Section of the Moon through the Centre ; and the jioon's atmos- phere confined to tho concave side. 6. Plan of the Moon's surface on the side opposite to the Earth. APPENDIX. ROTATION OF THE EARTH. Centripetal Attraction of Terrestrial Gravitation compounded with the tangential motion. The resultant or active force, as pressure, sustaining the rotation ; shown by the curves which indicate the