IMAGE EVALUATION TEST TARGET (MT-S) 1.0 1.1 1.25 111 u ■40 Fholograiidc Sciences Coiporation 23 WIST MAM STMiT VVnSTn,M.Y. 145M (716)«72-4S03 CIHM/ICMH Microfiche Series. CIHIVI/iCIVIH Collection de microfiches. Canadian Inatituta for Historical ISIicroraproductions / Institut Canadian da microraproductions historiquas T«chnical and Bibliographic Notaa/Notas tachniquaa at bibiiographiquaa 1 ha Inatituta haa attamptad to obtain tha baat original copy avaiiabia for filming. Faaturaa of thia copy which may ba bibiicgraphicaiiy uniqua, which may altar any of tha imagaa in tha raproduction, or which may aignificantiy changa tha uaual mathod of filming, ara chackad balow. 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D D n [3 D x/ Coiourad pagaa/ Pagaa da coulaur Pagaa damagad/ Pagaa andommagtea Pagaa raatorad and/or iaminatad/ Pagaa raataurAaa at/ou pallicultea Pagaa diacolourad, atainad or foxad/ Pagaa dteolortea, tachattea ou piquAaa Pagaa datachad/ Pagaa dAtachtea Showthrough/ Tranaparanca Quality of print variaa/ QualitA in^gaia da I'impraaaion Inciudaa aupplamantary matarial/ Comprand du material auppMmantaira Only adition avaliabia/ Saula Mition diaponibia Pagaa wholly or partially obacurad by arrata alipa, tiaauaa, ate, hava baan rafllmad to anaura tha baat poaaibia imaga/ Laa pagaa totalamant ou partiailamant obacurciaa par un fauiliat d'arrata, una palura, ate, ont ito filmtea A nouvaau da facon A obtanir la maillaura imaga poaaibia. Thia itam ia filmad at tha ratiuction ratio chackad balow/ Ca documant aat film* au taux da rMuction indlquA ci-daaaoua. 10X 14X 1SX 22X 26X 30X J 12X 16X 20X 24X 28X 32X mammKmm^^^ wm mmmmm trnf^^m mmmmmKummmmmm Th« copy film«d h«r« hM bMn r«produe«d thank* to tho gonorotity of: ThtNoviSNlia Ugialirtivt Library L'oxomplairo f Ilm4 f ut roproduit grico A la g4n4roait4 da: The Nova Seotia Lagithrtiva Library Th* imagaa appaaring hara ara tha baat quality poaaibia coneidaring tha condition and iagibility of tha original copy and in Icaaping wKh tha filming contract apacificationa. Laa imagaa auivantaa ont *t4 rapk oduitaa avac la plua grand aoin, compta tanu da la condition at da la nattat* da I'axamplaira f ilm«, at an conformity avac laa conditiona du contrat da filmaga. Original copiaa in printad papar covara ara filmad baginning with tha front covar and anding on tha laat paga with a printad or illuatratad impraa- aion, or tha bacic covar whan approprlata. All othar original copiaa ara filmad baginning en tha firat paga wrth a printad or illuatratad impraa* aion, and anding on tha laat paga with a printad or illuatratad impraaaion. Laa axamplairaa originaux dont la couvartura an papiar aat ImprimAa aont filmAa an comman9ant par la pramiar plat at an tarminant aoit par la darnlAra paga qui comporta una amprainta d'impraaaion ou d'iiluatration, aoit par la aacond plat, aalon la caa. Toua laa autraa axamplairaa originaux aont fllmte an commandant par la pramlAra paga qui comporta una amprainta d'impraaaion ou d'iiluatration at an tarminant par la darniAra paga qui comporta una talla amprainta. Tha laat racordad frama on aach microfiche ahaii contain tha aymboi — ^> (moaning "CON- TINUED"), or tha aymboi V (maaning "END"), whichavar appliaa. Un daa aymbolaa auivanta apparattra aur la darnlAra imaga da chaqua microficha, aalon la caa: la aymbola — »* aignlfia "A SUiVRE", la aymbola y aignlfia "FIN". Mapa. piataa, charta, ate, may ba filmad at difforant reduction ratioa. Thoaa too iarga to ba antiraly included in ona axpoaura ara filmad baginning in tha uppar laft hand cornar, iaft to right and top to bottom, aa many framaa aa raquirad. Tha following diagrama iliuatrata tha mathod: Laa cartaa, planchas, tableaux, ate, pauvant Atra fiimte A daa taux da reduction diff Aranta. Loraqua la document aat trop grand pour Atra reproduit en un aaui clichA, ii eat filmA A partir da i'angia aupArieur gauche, de gauche A droite, et de haut en baa, en prenant 'a nombra d'imagea nAceaaaire. Lea diagrammea suivant* iliuatrant la mAthoda. 1 2 3 1 2 3 4 S 6 ■.'«af^ o'ii I From the PniiiOCOPHiOAt. MAOAitirB for SeptemlMr 1893. 4^ 3ft* o>> -;*•'*'■ On tlie Hypotheses of Dynamics. By Prof. J, G. MacGrbgor, D.Sc, Dalhousie College, Halifax, N. S. PROFESSOR LODGE'S paper on the Foundations of Dynamics*, in which he criticises an Address of mine en the same subject t and replies to criticisms I have made J on a series of papers by him in this Magazine §, contains so much debatable matter that it would require more space than is available to give it full discussion. There are some points, however, which are of so much importance in the clearing up or our conceptions of the fundamental assumptions of Dynamics, that I venture in as brief a manner as possible to draw attention to them. (1) The Relativity of the First and Second Latvs of Motion. Prof. Lodge completely misunderstands the objection which was urged in my Address against the usual statement of the first and second laws of motion, and which had been pre- viously urged by various writers |l. He states it to be " that * Phil. Mag. current volume, p. 1. t Trans. Roy. Soc. Canada, vol. x. (1892), sec. iii. p. 3. X Phil. Mag. vol. xxxv. (1803) p. 134. § Vols. viii. (1870) p. 277, xi. (1881) pp. 36 & 529, xix. (1885) p. 482. II The list of writers which Prof. Lodge gives is obviously not intended to be complete. It omits C. Neumann ( Ueber die Principten der Galilei- Newtonschen Theorie, Leipzig, 1870), Prof. J. Thomson (Proc. R. S. Edin. vol. xii. pp. 568 & 730), Prof. Tait (ibid. p. 743 and ' Properties of Matter,' 1885, p. 92), H. Streintz {Die phj/tikalitchen Qrundlagen der Mechanik, Phil. Mag. S. 5. Vol. 36. No. 220. Sept. 1893. R I / 234 (A Prof. J. G. Mat'Grofior on the uniform motion is iinintelli^iMe or meaningless, unless you specify its direetion and velocity with refer( iiCe to a set of axes," whereas tlie real oljjection is that the laws themselves, in their usual form, are unintelligihle, unh'ss the axes are specified, to which the uniform uiotion or acceleration men- tioned in them is referred. His criticism is therefore neces- sarily somewhat wide of the m;irk. It may he sunuuarized thus: — (1) Uniform motion is perfectly intelligible; and therefore uo specification of axes is necessary in the enun- ciation of the first law. (2) The difficulties in the way of spo'jifying axes are practically insurmounta})le. With regard to (1), it will bo noted that it rests entirely on the intelligibility of uniform motion, and does not therefore touch the necessity of the specification of axes in the case of the second law * or of the first law in the form which Prof. Lodge has given it himself: — "Without force there can be no acceleration of matter "f. For in neither case is there any reference to uniform motion. With regard to the intelligibility of uniform motion, while it cannot be admitted that " such notions as axes of refiirenco are not at all necessary for the appn^hension of what is meant by a uniform velocity " (seeing that a uniform velocity is one whose magnitude and direction do not change relatively to the axes employed in its specification), it is nevertheless obvious that the specification of particuhir axes is not neces- sary for this purpose. But the intelligibility of the first law requires more than the mere apprehension of what is meant by uniformity of velocity. For it is not a mere statement ahovt uniform velocity, but an assertion that a })article in given circumstances must have a uniform velocity. Now a velocity which is uniform with respect to one set of axes may be variable with respect to others. It is therefore at once obvious that, if we employ the ordinary conception of force, the assertion which the law makes cannot hold for all axes, and consequently can have no definite meaning, unless i ■ft r Leipzig, 1883), L. Lange {Ber. d. K. Sachs. Ges. d. Wiss. zu Leipzuf, Math.- pJujs. Classe, Bd. xxxvii. 1885, p. 38.'^, and Die geschichtliche Entwickelung des Beti-egungsbegriffes, Leipzig, 188fi), and Muirhead (Phil. Mag, [5] vol. xxiii. 1887, p. 473), the last, however, being mentiuned subsequently in a footnote. * Mach, Streintz, Lange, and other German writers refer to the rela- tivity of the first law merely, because they employ as second law Galileo's law of the " phyeical independence of forces ' (Unabhangigkeitsprincip). The second law to which I refer is Newton's second law. t ' Natiire,' toI. xlviii. p. 62. I t^-z^'-^ •f lli/potheses of Dynamics. 2;]5 wo are told what are tlio axes by reference to vvliich it iloes hold*. Mucli may of course be derived from the first and second laws without specification of axes. The whole science of dynamics bears witness to that fact. But, as Streintz has shown in the work referred to above, much practical incon- venience and much unnecessary complication have arisen from the employment of these laws in their vague form ; and I shall have occasion to refer below to one paradox, the absoluteness of rotation notwithstanding the relativity of motion, which receives its solution when the relativity of these laws is recogn'zed. The specification of axes by reference to which the first and second laws hold, or of what may be called dynamical reference systems, is thus no mere refinenuMit of the pedantic mathematical mind. On the contrar}'', it satisfies a felt want. The want is not felt indeed in dealing with the simple problems of the common school. For the rough experiments which are usually cited in elenunitary text-books as suggesting the laws show that it is by reference to axes fixed in the earth that they ar<^ supposed to hold ; and this tacit specafication is quite sufficient for the discussion, e. g., of the inclined plane and the wheel and axle. But when we comk; to treat the problems of theoretical Astronomy, it is at once obvious that we cannot assume the laws to hold with respect to these axes; and th«; question forces itself upon the attention : What are the axes by reference to which they must now be considered to hold ? And the question having been raised must be answered. The critical student who has seen in his study of kinematics that velocity and acceleration are relative con- ce{)tions, will not be convinced by Prof. Lodge's "opprobrious or perhaps complimentary epithets " that they lose their relativity when applied to the motion of bodies. Turning now to the second criticism, it is obvious that to one who thinks it is proposed to specify axes by means of which the magnitudes and directions of velocities may be described absolutely (p. 8), the difficulties in the way must appear * Should Prof. Lodge, therefore, endeavour to crush a doubter of the first law, as he tells us he would, by saying to him : — " If the speed and direction of a freely moving body vary they must vary in some definite manner; very well, tell me in what manner they are varying. You cannot, unless you can show me absolutely fixed lines of reference,'' — the doubter need be at no loss for a reply. He has but to say : — If you will kindly tell me what are the axes by reference to which you hold the velocity of the body to be uniform, I will then tell you how I suppose it to vary. But unless we agree upon axes of reference it is impossiWe for UB to compare our respective axioms. R2 2dG Prof, J. G. MacGrogor on the insuperable*. They would be in that case the difficulties attending the solution of an inconceivable problem. That the actual problem has only recently been attacked is due not so much to its difficulty as to the fact that the necessity of its solution has boen apparent only since the full recognition of the essential relativity of velocity and acceleration, whether uniform or variable. That there are difficulties, however, is obvious from the fact that only some of the methods emploved appear to be sound, and that a number of writers have attacked tne problem and left it only half solved t- What the difficulties are may be shown best by a sketch of the efforts made to overcome them. There would seem to be two legitimate ways of finding dynamical reference systems: — (1) by re-studying the experi- mental results for the deduction of which the laws of motion were enunciated, and re-formulating these laws ; and (2) by assuming that, since the laws of motion in their vague form have been abundantly tested in the hands of men enabled by a kind of dynamical instinct to use them aright, there must be axes by reference to which they hold, and proceeding to determine these axes by the aid of the laws themselves. The former method, the historical-critical, is that employed by Prof. MachJ. He points out that Galilei observed the first law to hold, by reference to points fixed in the earth, for motions on the earth's surface of small duration and extent, and that, when Newton came to apply it to bodies moving in space, he generalized it, showing that, so far as could be determined, it held for the motions of the planets by reference to the distant and to all appearance relatively fixed celestial bodies. And he holds that the first law, when referred, so far as space is conc.rned, to the fixed stars, and, so far as time is concerned, to the earth's rotation, is to be regarded as a sufficient approximation to accuracy for practical purposes, and as forming as close an approximation as it will be possible to obtain until a considerable widening of our experience occurs. It seems to me that the historical-critical method might carry us farther than this. For we now know that the so-called fixed stars are not fixed ; and means have been devised of correcting observations made on this assumption. We also know that the laws of motion do not hold when referred to a time-scale determined by the earth's rotation ; and a rough correction has been determined for application * The fact that Prof. Lodge regards motion with respect to the sether as absolute motion (p. 30) perhaps renders this statement doubtful, t Neumann, J. Thomson, and Muirhead. See works cited above. X Die Mechnvih in ihrer Entwickelung \ Leipzig, 1889, pp. 217 & 481. ;, Jli/fot/u'Si's of Dynamics. 237 to tin's time-scalo, in tlio case of motions oxtending over long periods of time. The first law, when expressed by reference to the fixed stars and the earth's rotation, is therefore no longer regarded as sufficiently accurate for all purposes ; and the exact expression of the law, as empirically determined and employed in actual work, changes from day to day or at least from decade to decade. The question therefore arises : Can we put the laws of motion into g(?neral forms such that the empirical forms which they may have at any time may 1)0 regarded as special cases determined by the state of know- Icdire of the time ? ■pi flV The latter of the two methods* referred to above is intended * Macli, though hokling, as seen above, to the empirical result of the historical-critical mothol, jrives in liis Mechunik (p. 218) au interesting " remnant," as he calls it, of liis efforts to apply the second method. He h»)lds tliat, in using the first law in its Newtonian I'orm, we may he regarded as employing the universe, or a sufficiently large portion of it, as our reference system, and on the following grounds: — "Instead of saying the velocity of a mass ^i remains constant in space we may also employ the expression, the mean acceleration of the mass /i, relatively to the masses m, r«', &c., at tlie distances ;•, r', &c., is zero, or de2m~' The latter expression is equivalent to the former, provided we take into consideration a sufficient number of sufficiently distant and great masses, the mutual influence of the nearer small masses being m that case negligible." If this be so, the first law may b« expressed as follows : — The mean acceleration of any particle, relatively to the other particles of the universe, or of a sufficient portion of the universe, is zero, provided the particle is not acted upon oy force, — an expression which obviously has not the same vagueness as the Newtonian form of the law, though practically, as Mach points out, if is not more readily applicable, on account of the impossibility of making the summation necessary for the determination of the mean acceleration. How this result is arrived at, Maeh does not say. But it is easy to prove it to be one of the properties of the centre of mass, that the com- ponent acceleration, in any direction, of any one particle of a system, relatively to the centre of mass ol' the system, is equal to the mean com- ponent acceleration of this particle, in the same direction, relatively to all the other particles of the system, provided the mass of the particle is small compared with the mass of the system. In making the above statement, therefore, Mach would seem to assume that the uniform velocity contemplated in Newton's form of the first law is a velocity which is uniform relatively to the centre of mass of the universe, or of a sufficiently large portion of it ; and if that be so, he assumes a partial specification of a dynamical reference system. It would also appear that the portion of the universe taken into consideration need not consist of numerous and distant particles, but must simply have sufficient mass. It is obvious that if the above assumption be made, not only may the first law be thrown into the above form, but also the second law may be thrown into a corresponding form. The making of this assumption, how- ever, introduces a complication. If we assume merely that there are axes 238 Prof. J. G. MuoGregor on the to give laws of tliis kind. Prof. James Tlionuoii may bo snid to have einfiloycd it when he showed how, hy observation of successive rehiiive j)ositions of partiehvs given as moving in straight lines, tlio iix(s hy reference to whicli their path.s are rectilinear may he determined geom<'trically*. Thomson and Tiiit may bo said to emjiloy it also, when tliey show, by a d(Mluction from the first law, how we may imagim? ourselves as obtaining " fix(!d directions of reference'' t« l^ut these authors make no attempt to give a formal sj)ecificatioii of a dynamical refenmoe system. Lang(^ employed this method in the pa])er refc^rred to above, basing liis suggc^stion as to specification on a kinematical result, viz. that for three, or fewer than three, points, which are moving relatively to on(^ another in any way whatever, it is always possibhi to find a system of coordinates, indeed an infinite number of such, systems, by reference to wliich these points will have rectilinear paths ; while for more than three such points this is possible only in special circumstances. It follows that the law of the uniformity of the direction of motion of jiartides free from the action of force is, for three such particles, a mere convention, and that it is a result of experience only in so far as it applies to more than three particles l)y reference to one and the same system. Hence just as the dynamical time-scale is defined as a time-scale by reference to which a particle free from the action of force moves with a uniform speed, so the dynamical reference system may be defined as a system by reference to which three particles free from the action of force move in rectilinear paths. Following out these considerations he finally proposes to enunciate the first law in the following form : — Relatively to any system of coordinates by reference to which three particles projected from the same point in space and thereafter r by ret'erencp to wliich the first and second laws hold, it may be proved by means of the second and third laws that relatively to" these axes the centr»> of mass of a system of particles will have no acceleration, provided no external forces act on the system. While, therelure, the iissumptiou that the centre of mass of the universe may be em])loved as thw origin of a dynamical reference system is justified, it is obvious (I) that if, iu employin}^ Mnch'a expression of the first law, we restrict ourselves to a part of the universe, it must be a uart on which no external forces act ; and (2) that since, in obtainino- this form of the law and the corresponding form of the second law, we employ the third law of motion, the new laws are not merely new expressions of the old laws, but involve the (hird law in addition. * See also I'rof. Tait's solution of this problem by Quaternions in the paper «'ited above, t 'Treatise ou Natural rhilosophy,' vol. i. pari 1 (1879), 5 24ft. ^^ l^-> Jfi/j>i)lhexen of hynamics. 239 left freo from llio uction of foroo, which do not, however, lio in u straight line, descrih(^ any thn'( -trai;^ht lines inter.sect- injr in a point (the axes of coordinates for (example), the path of any fonrth particle free from foree will he rectilinear. And relatively to any tinie-scah^, hy rctercnce to which one particle W'K^e from force will, wlien its motion is n^ferred to the ahovo axes, move with uniform speed, every other particle free from force will move with uniform sp»!ed, if its motion he referred to the same axes. It was this UKithod also which I employed in my Address, when I h;id not yet met with iiange's paper, the conclusion nniched heing, that the 1st and '2,n(\ hiws hold relatively to any particle not act(ul upon hy force, as point of reterence, and to lines drawn from it to other particles which are unacted on hy force and liav! of a non-rigorous proof (p. 10). I " That Newton really regarded himself as having deduced the third law from the first is rendered extremely doubtful by the fact that he retained this law as one of his axioms. But it seems clear that he re- garded part of what we now consider to be included in the third law to be capaDle of deduction. That Newton regarded the third law as less general in its applicability as an axiom than we do may be gathered from his comments on it. lie illustrates it by reference to the finger pressing a stone, a horse hauling a stone by means of a rope, and bodies impinging upon one another, — all cases of palpably contact-actions. And he concludes his illustrative comments by saying : — " This law holds also in cases of attraction, as will be proved in the following Scholium." The fact that his third law states action and reaction to be equal and oppssite but says nothing as to their being in the same straight line, forms corro- borative evidence that he regarded his law as applicable directly to con- tact-actions only. For in such actions it would follow, from the opposition of action and reaction, that they must be in the same straight line. It would thus appear that Newton regarded the application of the third law to attractions as capable of deduction." — My Address, p. 10. § " In attractionibus rem sic breviter ostendo. Corporibus duobus quibusvis A, B se mutuo trahentibus, concipe obstaculum quodvis inter- poni, quo congressus eorum impediatur. Si corpus alterutrum A magis trahitur versus corpus alterum B, quam illud alterum B in prius A, Hypotheses of Dynamics. 245 out in detail. If I understnnd it aright (it is so condensed as to be somewhat obscure) it is as follows : — Let A and B be two mutually attracting bodies, and let F and F' be the attractions on A and B respectively. Imagine any obstacle, O, interposed between them so as to prevent their approach. Then, provided the attractionul stress between A and B be independent of the existence of other stresses between them and other bodies (assumption No. 1, the " physical in- dependence " of stresses), F and F' will still be the attractions on A and B respectively. Let R and R' be the action on A and the reaction on 0, respectively, of the contact stress between A and ; and let Rj and R/ be the corresponding forces for B and respectively. Since there are no external forces acting on the system of A, B, and 0, it follows from the law of the conservation of the motion of the centre of mass* (assumption No. 2, which we may call the generalized first law of motion), that the centre of mass of A, B, and will move uniformly. If A, B, and be rigid bodies (this restricts the argument to the case of attracting bodies kept at a constant distance from one another) 0*s motion will also be uniform. Hence the resultant force on must, by the first law of motion, which is a particular case of the generalized first law, be zero. But, by the law of the composition of forces, which is a deduction from the second law of motion (assumption No. 3), this resultant force is R' + lVf. Hence R'= — Ri'. Now, by the third law of motion regarded as applicable to contact stresses (assumption No. 4) we have R= — R' and Ri= — R/. Hence R= — Rj. But the motions of A and B must be uniform for the same reason as that of 0. Hence by the first and second laws as above, F + R=0 and F' + Ri = 0. Hence also F=— F'.— If thi^; is a correct statement of Newton's argum(mt it is obvious that it does not make the deduction which is claimed for it | . obstaculum magis urgebitur pressione ccn-poris A quam pressione corporis B ; proindeque non iiianebit in aequilibno. Praevalebit pressio fortior, facietque ut systema corporum duorum et obstaculi moveatur in directum in partes versus B, motuque in spatiis liberis semper accelerate abeat in infinitum. Quod est absurdum et legi primte contrarium. Nam per legem primam debebit systema perseverare in statu suo quiescendi vel movendi uniformiter in directum, proindeque corpora seoualiter urgebunt obsta- culum, et idcirco sequaliter trahentur in invicem. — Principia : Scholium to A.xiomata. * Newton had previously (Cor. 4 to Axiomata) proved this law, assum- ing, in the proof, the third law as applicable to all stresses. t I assume, as Newton does, for simplicity, that the forces are all in one straight line. X See Lange, Beivegungshegriff, p. 57. 246 Prof. J. (I. MiicGregor on the MaxwelP.s vorsion of Newton's iir«5uinont*, which ho regards as " a deduction of the third hiw of motion from tlie first/' may, as I showed in my Address (p. 12), be attacked on two grounds. First, it assumes that the attraction between the mountain and the remainder of the earth ''s the only stress between them, ignoring tlie stress at their surface of contact, an inequality in the action and reaction of which might ob- viously neutralize the " residual force " due to the assumed inequality in the action and reaction of the attraction. Secondly, the conclusion, thus illogically obtained, is not the third law of motion. For the former asserts the equality and opposition of the action and reaction of the stress between two parts of a body, to which body, as a whole, the first law has been assumed to apply, while the latter makes the same assertion for two bodies, to each of which the fiist law is applicable. That this criticism is sound becomes especially apparent, if we refiect that when dealing with rotation and strain we must regard the laws of motion as applicabl' to particles or elements, the first and second laws being held to apply to each particle, and the third law to the stresses between pairs of particles. Maxwell's deduced law would apply only to the actions and reactions of the stresses between the parts of single particles which, as Prof. Lodge says (p. 11), are "not worth troubling about." It would tell us nothing about the stresses between pairs of particles, and would thus be of no use in the solution of dynamical problems. Prof. Lodge's version of Newton's argument : — " Jam the bodies apart with a rigid obstacle, then you have reduced their action to contact action " &c., (p. 10), is so condensed that it is hard to analyse. But it is easy to see that its first statement is incorrect. For when we "jam the bodies apart " we do not reduce their action to contact action. The attraction continues. We have simply introduced, in addi- tion, two contact stresses. The premisses, therefore, being thus fau)ty, the conclusion cannot be warranted. Prof. Lodge appears to have abandoned the deduction of the third law from the first in its usual form. " Whether," he says, " it is deducible from the first law or not niay be • " If the attraction of any part of the earth, say, a mountain, upon the remainder of the earth, were greater or less than that of the re- mainder of the earth upon the mountain, there would be a residual force acting upon the system of the earth and the mountain as a whole, which would cause it to move off with an ever-increasing velocity through in- finite space. This is contrary to the first law of motion, which asserts that a body does not change its state of motion unless acted upon by external force " (* Matter and Motion ', Arts. Ivii. and Iviii.). s Hypotheses of Dynamics. Ul held to depend on how j^enenil the terms uro in which that law has been stated. If it can be uxioniatically assert(Hl that the centre of nass of a rigid tofstem moves uniformly until an external force acts on tlie system, and also that the system does not be[;in to spin, then the third law is established. For since zero acceleration means zero force, it follows that all the internal forces add uj) to zero, and have no moment ; and since the system can be dissected bit by bit without ceasin*; to be a system within the scope of the first law, it follows that no stress can contuin an unbalanced force or couple." Here, then, we have a new deduction, on which I would nuiko two ren^arks : — (1) The conclusion is obviously too general. For since the assumptions specified are made for a rigid system only, the " no stress " of the conclusion should clearly be — no stress between the parts of a rigid system. The con- clusion would thus become only a [)articular case of the third law. (2) That even this modified conclusion cannot be ob- tained without additional assumptions, and, even with them, by the method of dissection, may readily be shown. *' All the internal forces add up to zero and have no moment." How do we know this? Only by the aid of familiar deduc- tions from the second law of motion. Thus the second law is assumed. Dissect away one })article from the system. By the second law, as above, the internal forces of the re- maining particles now add up to zero and have no moment. But we cannot assert this to have been true before the removal of the particle, unless we assume the physical inde- pendence of stresses. If this second additional assumption be made, though we now know that the actions and reactions of the stresses between any one particle and the remaining particles add up to zero and have no moment, we cannot con- clude that "no stress can contain an unbalanced force or couple," because they would add up to zero and would have no moment, also, provided any inequality in the action and reaction of one stress, and their resultant moment, were neu- tralized by inequalities in the actions and reactions of other stresses and by their resultant moment, respectively. While Prof. Lodge's method of dissection will not give even the modified conclusion, even with the aid of the above additional axioms, the reverse process will give it without his assumption as to spin. For in a system of two particles the conservation of motion of the centre of mass and the second law together tell us that the action and reaction of the single stress in the system are equal and opposite. And in a system of any number of particles, the axiom of the physical independence of stresses tells us that the stress between any two particles is the same as if there were no 248 Prof. J. G. MacGregor on the I others acting, and that, therefore, in the case of all the stresses action and reaction are equal and opposite. It is at once obvious that this argument will hold also, whether we restrict ourselves to rigid systems or not. Prof. Lodge does so, probably because he feels he cannot appeal to the experience of the human race with regard to the motion of the centre of mass of a non-rigid system. Had he adopted as his axiom the generalized first law (which he would be justified in doing according to my conception of an axiom as a proposition by means of which it is found possible to co- ordinate dynamical phenomena generally), then, with the aid of the second law and the physical independence of stresses, he might have deduced the equality and opposition of the action and reaction of all stresses *. But even then he would have deduced only what is explicitly stated in the third law and not tho whole law. For just as the second law, by the generality of its assertion, implies the " physical indepen- dence of forces," so the third law implies the physical independence of stresses, at least so far a3 the equality and opposition of their action and reaction are concerned. This implied part of the third law is assumed in the above deduction. So much for the asserted possibility of deducing the third law from the first. Prof. Lodge has held also that it may be deduced from, the second f. Divested of its " muscles and clothing," his argument is as follows : — Action may be taken to mean simply the whole force applied to the body con- sidered. The reaction of a body is defined as equal to the product of its mass into its acceleration. The second law of motion may be expressed in an equation on the one side of which we nave the resultant force on a body or the action, on the other side of which we have its mass multiplied by its acceleration, which we have agreed to call its reaction. Thus action is equal to reaction. After a few paragraphs of explanatory matter he changes the expression of this result, without any attempt at justification, to the following ; — The reaction or mass acceleration of a body is equal and opposite to the resultant of all the forces acting on it. It is hardly necessary to discuss this argument. It will be suflSeiently obvious that if the definitions of action and reaction be ac- cepted, reaction, if assumed to have direction, must be co-directional with action, not opposite to it, and that there- * Streintz (loc. cit. p. 131) and Muirhead {loc. cit. p. 477) point out the possibility of deducing the third law from the generalized first law, but do not perceive the necessity of assuming the physical independence t 'The Engineer,' vol. lix. (1885), pp. 217, 311, 380. l< Hypotheses of Dynamics. 249 fore the conclusion which ought to be (h'avvn is not even expressed in the same words as the third law. It is also just as obvious that even if the conclusion drawn had been war- ranted, though expressed in the same words as the third law it would not have been the same law, because the term reaction would be used in entirely different senses in the two laws. That the above efforts to deduce the third law from the first or second should thus prove futile need not surprise us. For the second law gives us a quantitative statement as to the effect which is produced in a particle by a force; while the third tells us that forces always occur as one-sided aspects of stresses, and gives us the relation between the two forces t^ of which every stress consists. Had these laws been recog- nized as being thus complementary to one another, efforts to deduce either from the other would have been seen before- hand to be doomed to failure *, and the above dreary refutations would not have been called for. (4) Prof. Lodge 8 Deduction of his Line of Conservation. Though Prof. Lodge still holds (pp. 11 & 14) that the conservation of energy (as defined by him) " can be deduced from Newton's third law and fi-om the denial of action at a distJhice," and indeed gives a new version of this deduction, he admits that his deduction applies only to conservation during transfer, and that conservation during residence or " storage '* is incapable of deduction f. How he reconciles * This seems, at first sigbt, not to agree with what Mach says (Mechantk, p. 228) after having referred to the subject matter of New- I ton's first and second laws, viz. : — " Tlie third law contains apparently j \Aomething new. We have already seen, however, that without the cor- I rect conception of mass it is unintelligible, and that on the other hand, I through the conception of mass, which itself can be obtained only ( through dynamical experiences, it is rendered unnecessary." As, accor- j ding to the ordinary interpretation, the idea of mass is given in terms of i force by the second law, Mach would seem to hold that the third law is not independent of the second. This is not tlie case, however. Mach had previously shown that if we interpret Newton's second law by the aid of his definitions, this law does not give us a clear conception of mass. He himself obtains the conception, without reference to force, by an appeal to experience, which takes the place of the appeal made in the third law and thus renders it unnecessary. t After dismissing, with some hesitation, the " plausible " method of establishing a law of nature by appeal to definition, he suggests that conservation during storage should be adopted as an axiom ; but he does not meet the argument given in my Address to show, that if we retain Newton's laws, it is illogical to employ the law of conservation as an axiom, and that if we adopt the latter la\^ as an axiom, Tait's suggestion (Ency. Bn$., Art. Mechanics, § 299) is the only 'ogical one, viz., that Newton's laws should bf abandoned and the law of transference of energv adopted in.*tead, r/iil. Mag. S. b. Vol. 36. No. 220. >iept. 1893. S 2r)0 Prof. .1. (i. Mii('(j}r(»o()r on f/ic the reiteration of his oM claiiu witli this uchnission we are not told. With regard to my crltieisni of his earliest mode of making this deduction, he replies tluit the ajjpcal to experience which I pointed out as havin(( Immmi made in his arjLfument was a mere piece of politeness, whicli mi^ht have heen omitted without affecting the argument. 1 tiiink if ho will look into the matter he will find that if he had heen less polite his reasoning would have heen faulty. It is imnecessary to occnpy space in proving this, howev<'r, hecause the new de- duction, given in the pr(!sent paper and referred to helow, emhodies exactly the same fallacy as the old one. In reply to my criticism of the law of conservation de- duced hy the argument of his third paper ((piite a different law as I pointed out from that obtained in the earlier paper, though Prof. Lodge does not seem to realize this), viz,, that it was of the same natures as the law of the conservation of momentum, his energy as defined in that })aper being con- stant in quantity, because equal quantities of positiv^e and negative energy must always be produced together, he states that his law is deduced from a less obvious assumption than the conservation of momentum. This is |)ossibly true ; but it does not affect the natun; of the law deduced. The law of the conservation of electrical quantity is obtained in a dif- ferent way from either, but is nevertheless a law of the same kind. The new version of the deduction of the conservation of energy from the third law and the assumption of contact- action, is based upon a new definition of energy as " the result of work done," or " the result of activity lasting a finite time." As tliis is rather vague, work done on a body having a variety of results, Prof. Lodge proceeds to expound his definition and tells us that energy " is a name for the line-integral of a force, considered as a quantity that can be stored " *. Here, again, is the appeal to experience, — that the line-integral of force may be considered as a quantity which can be stored. If it is introduced merely out of politeness, it must not be used in the argument. If used in the argument, it forms an unacknowledged assumption. The argument is as follows : — " Bodies can only act on one another while in contact, hence if they move they must move over the same distance ; but their action consists of a pair of equal opposite forces ; therefore the works they do, or their activities, are equal and opposite ; therefore, by defi- * His comments show that he should have added here the words " in a. body." But this does not affect the preseut argument. IJypothenes of Vy mimics, 251 nition, wlmtcvor onor^y the ono loses the otlior gains. In other words : in nil oases of activity, enor«;y i?* simply trans- ferred from one body to another, without alteration in quantity.'* Now itisohvious that we ennnot pa^s by dejinition from work done or activity to energy, for energy has been defined as the result of work done or activity, the line-integral of a force considered as a quantity which can he stored. We must first pass from the e(iualitv and oj)position of the works done (or line-integrals of the forces) to the equality and opposition of the results of the work* done (or of the line- integrals of the forces considered as (/nant/'ties irhich can he Ktored). How is this passage made? We are not told. But it is obviously by the assum|)tion that work done on a body is equal to the result of work done, or that the line-integral of a forc(» may he considered as a (juantity which can be stored. And as obviously, this assumption is the law of the conservation of energy. Thus the conclusion of the argu- ment, which, as in former deductions, is clearly conservation durinfi transfereni^e onlv, is obtained bv assuming the law of conservation generally. It would indeed be a remarkable thing if it were possible, in the case of systems whose parts act upon one another only when in contact or at constant distance, to deduce the con- servation of energy during transference from the third law of motion alone. When wo make no assumption as to the distance at which action may occur, wo require, in order to obtain the law of transference, to obtain first the general law of conservation, for which purpose we have to assume the second law of motion, aiid some such axiom as the impossi- bility of the perpetual motion. Having obtained tVom these axioms the general law of conservation, the third law then gives us the law of transference. Why, then, when we restrict our attention to systems exhibiting constant distance action * only, should it be possible to deduce the law of transference independently of the law of conservation? This is a logical question to which it should be possible to give a clear answer, if it is possible to make the deduction referred to. (5) Prof. Lodge's J)ednctinn of Contact -action. In former papers t Prof. Lodge claimed to prove the * Prof. Lodge's argument assumes constant distance action, not spe- cifically contact-action ; for " if they move they must move over the same distance" is true of actions at all constant distances, not merely of actions at distance zero. t Phil. Mag. [5] viii. p. 279, xi. p. ;^0. S2 252 Prof. J. G. Mac'(Jivgor on the incompatibility of action at a distance {\) with the law of the conservation of ;«[)Pr.) In n't'orciicc to his Htaicmciit that his law is as axiomatic as tho ordinary law, I showcj that tho latti«r is tho more goiKU'al ill its applicahility to (lynainical |)rol)U'ins. l*rof. Lo(l;r(. (loclarcs hmiscU" to Ik; in cuiiro a;i;rit an opposing force, and energy being defined as power of doing work f, the ordinary law asserts that energy is conserved. It is sometimes expressed in terms of tho fiction of action at a distance and sometimes also in terms * It is this confusion, I tliink, wliicli has led Prof. XjuA^g, in his com- parison of our ivspeiHive types of mind, to niiike tho ontirely erroneoua stiitenieut that 1 am willing '* to baso Physics on action at a distance " (p. 2, footnoti)). To it is duo also the stati'UU'nt of p. 10 in which the ordinary conception of potential enur^'y is ascribed to " the believer in action at a distance.' ' t Prof. Lodge's extraordinary objections to this delinition are easily met. — (1) It is '' vague.' Doubtless it is to om^ who can make the state- ments (juoted below. Compare its precision with that of tlie detinitioua by which it is to be replaced: — -"etlijct of work done; " ''result of work done ; " " line-integral of a force considered as a (|ua!itity whieh (!an be stored." The formal definition of his * Mechanics' : — " Energy is that part of the elVect produced when work is done upon nnitter, wlucii is not an accidental concomitant, but really owes its origin to t!ie work, and could not, so far as we know, have Ijeen produced without it ; and which, moreover, confers upon the b uly ]>iis^Hssiiig it un increased power of doing work," — would seem to imply tiiat he rightly considers his (jwn delinition .>o vague as to rerpiire to be supplcnuuited by tho ordinary definition. (2) '* Plenty of energy has no power of doing work, at least, uo power that we can get hold of. " \or can it have according to the definition. Probably what is meant is that plenty of bodies possess energy which we cannot utilize ; but our ability to get hold of power is no criterion of its existence. (.'Ij "A given amount of energy may have an infinite working-power, since it can do work at every transfer without itself diminishing." As just stated, according to the definition, energy cannot be said to have any working-jwwer at all. It is the body or system of bodies possessing the energy »vhich has the power. (4) " It is bold to maintain the conservation of working-])ower in face of the doctrine of the dissipation of energy." The conservation of [tower of any kind is quite consistent with diminishing opportunity of e.vercising it. 254 Prof. .1. G. Muciircgor on the of the fiction * of contact-action, liut it may be cxpresMMl without any reference to s^uch fictions. Moreover, it nuiy he deduced from the second law of motion and th(* impossibility of the perpetual motion, neither of which axioms involves puch fictions. It is thus (|uife general, involvin;^ no assumption as to the distance at which hodii'S can actt)n one another, "nd applyin;; to all cases of action, whether at distance zero, ut coufiftant distance;, or at variable (li^tance. It will thus be evident why " the true law " does not lend itself to fictions, and why the ordinary law does. "The true law" does not, because it alreaily end)odies a fiction. The ordinary law does, because it endjodies ncne, and is e(|ually applicable, whatisver fiction we may find it convenient in the meantime to assume or may ultimately find aj)parently coincident with fact. It is no discredit to " the true law " not to lend itself to fictions, j)rovided the fiction it embodies assists us in coordinating the whole range of dynamical nhe- nomena f. But discredit must attach to it so long as tnere an; groups of phenomena to which the ordinary law can, while " th« true law '* cannot, be ap|)lied. It will also be evident that, sinc(i the ordinary law involves no fictions, there need he no fear lest the employment of it should lead to the confounding of fictions with realities. Not only does the ordinary law make no assumption as to the distance at which action nuiy occur, it also assumes noihing as to the mechanism of action, and holds whether bodies be regarded as acting on one another through a medium or not, and if they are, whatever the medium may be through which their action is supposed to bo conveyed. To speak of the law as on this account incomplete seems to me to bo incorrect. Until we find some hypothesis as to acting mechanism which will enable us to coordinate dyna- mical [)henomena, the science of dynamics must of course be incomplete ; and doubtless as soon as possible some such hypothesis should be framed. But no such axiom has yet been suggested which is capable of general application. We cannot therefore help ourselves. The foundations of dy- namics must in the meantime remain incomplete, though they are none the less firm on that account. Even, however, when the time of omissions and slurrings shall have passed, the law of the conservation of energy will be no more com- * As Prof. Riicker has pointed out in ' Nature,' vol. xlviii. p. ] 26, contnct-action is as inccacei-vable as action at a distance. Both are t lus equally fictions. t Prof. Lodge holds, somewhat inconsistently, that " in a fundamental or theoretical treatment convenient fictions are better avoided " (p. 17). il Hiljiothfuef of hynainics. r^h fl 'A f»l«'lo tlitiM it i?« now. \\v iiiav Imvo actiiiircl moio di^Hnito <'(»n('opti»ms as to tlw cliariictcr of (.•cr'ain tonus of ciK^r^^y, as to tli(» rjiodn ol' transt't'rciicif of lmum'^v, ami us tt) tlio phuto of rcsitlciicr of |M)t(Mitial (MK'r;;v : Imt as tlic law of coiisfrviitioii makes no .Htatcnient on siu-li (loint- wliicli will tlicrchy '>« <'()ni|»l<'t'.'(l, our prHsunt i^noiano*^ willi r(';,Mr(l to tluMU dooH not t'ltndur it incomplete. Wliilo in tiio event of some hypolla'sis as to uetin^^ inoclianism heeomin^' axiomatic tliert) wonid \w somo ro- adjustm(Mit in the oidintirv conception of potential <'i>cr;^y» tluiro would not he nearly so mui'ii as I'rof. Iiod;;e supposes; for thou;jjh iiis accoimt of this conception (p. 1(5) is obviously u l)Urles(|ue, he clearly does helieve that it involves an "erroneous localization of eiuM'ejy," that, <•.//., in the case of a raised stone, the potential ener;fy must he supposed, nearly till at any rate, to he residenr in tlu* stone*. Tins impression, however, may readily he sIk \vn to he <'rroueous. Tlie [)oten- tial energy (see the delinitions of work done and energy, given ahove) may he said to helone; to the system of earth and stone, because work may he done hy the earth or hy iho stone or hy hoth during the approach of these i)odies. That it cannot he said to belong to either, is obvious from the <'on- sideration that, if either beheld fixed relatively to a dynamical referenci^ system, the work done during approach is then (lone by the otlier. How much of the work done during approach is done by {\w one and how nuK-h by the other, when both an' in motion, de[tends upon tlu; forces against which they move and their respectiv*^ displacements ndative to sueii system. It is thus obvious that according to the ordinary conception we can assert no more than that the potential energy belongs to the system, that this conception theref'i>'*e involves no localization of the onergv in the svstem, and consequently no erroneous localization. Tliis of course arises from the fact that the ordinary con- ception of potential energy involves no assum|)fion as to acting mechanism. 8houKI some sutKciently dt?finite hypo- thesis of this kind become axiomatic, it would then become possible to localize potential energy. If, e.ci., we should come to hold that bodies ( onsist of rigid particles connected by, and acting on one another through, an elastic medium, it * He objects to tlio ordinary coiiceptiou of potential energy as being " a mere receptacle for stowing away any portion of energy wliicfi it is not convenient for tlie moiuent tu attend to," yet admits (p. 24) that his own potential energy belongr^ to the same " temporary order of ideas." He also defies " any one to realize it as a thing." If he will define " thing" we may ))erha]).s try. 256 Prof. J. G. MacGregor on the would then be obvious that the potential energy must be con- sidered to be resident in the medium. But until some such hypothesis becomes axiomatic, no localization of potential energy is possible. It should be noted here that the adoption of the hypothesis of contpct-uction alone does not enable us to localize potential energy. As an axiom of acting mechanism it is incomplete ; and it involves onlv the residence of energy in some body or other *. If we are to know in what body the energy resides, the axiom of acting mechanism must be made sufficiently complete. Thus, as just stated, if we assume, in addition to mere contact-action, that material bodies consist of rigid particles, and that the medium is elastic, the potential energy in the case of the raised stone must be considered to be resident in the medium. If, however, we assume bodies to consist of elastic particles, then the |)otential energy must be regarded as possibly resident partly in the medium and partly in the particles of the earth and stone. Prof. Lodge does not seem to realize this; for though he has proposed no hypothesis as to acting mechanism beyond that of contact-action, he has no hesitation in saying dogmatically (for he makes no attempt to justify the assertion) that in the case of the raised stone and in similar cases f the potential energy resides in the medium J. Prof. Lodge claims that the law of the conservation of contact-action energy is more precise and detinite than the ordinary law " because it is the law not only of conservation, but of identity.''' As to what is meant by its being a law of identity, he gives us two statements, of which we may con- sider the later first. " My proposiciou,'* he says, in his 1 / * I use tlie word body here in Prof. Lodge's general sense as applicable to a portion of the medium as well as to a material body. When, accord- ing to the contact-action conception, potential energy is regarded as resident in a body, the body must be considered to be an elastic body in a state of strain (p. 20). Since it is thus considered to consist of parts capable of relative motion, it is a system of relatively movable pi ts. Thus, according to Dr. Lodge's conception, potential energy is resident in systems just as truly as it is according to the ordinary conception, only the systems in the one case are small, while in the other they may be large. See Mr. E. T, Dixon'd letter to ' Nature,' vol. xlviii. p. 102, and Dr. Lodge's reply, p. 120. t Prof. Lodge is not so precise in his localization in all cases. In the case of the bent bow, even the r>rdinary conception of potential energy would admit of our localizing it " in the bow, and in the case of the gunpowder that conception would give us a more definite localization than " in the powder." X Probably my meaning in the statement quoted by Piof. Lodge on p. 16, footnote, will now be clear. Hypotheses of Dynamics. 257 {\ 'U latest paper (p. 30), " amounts to just this, that whatever energy appears in a bounded region must necessarily have passed through the boundary." It is quite obvious that the assumption of contact-action together with the law of the conservation of energy do justify this proposition. It is equally obvious that, with the ordinary conception of energy, this proposition cannot be asserted. It may hold, but we cannot assert that it does. Why this difference ? Not because the law of conservation is more precise or definite in the one case than in the other, but because we have a more complete conception of transference. With the ordinary conception of energy the only source of knowledge of trans- ference is the third law of motion. With the other conception, we have both the third law and the axiom of contact-action *; and it is because of the greater definiteness which this latter axiom gives to our conception of transference, that it enables us to assert that if energy appears within a bounded region, it must have been either conveyed or transferred across the boundary f. The above proposition seems to Prof. Lodge " to confer upon energy the same kind of identity or continuous exist- ence (or, if you please, objectivity) as matter possesses " What kind of identitv or continuous existence matter is supposed to possess (we need not refer to anything so meta- physical as objectivity) may bi; gathered from the earlier of the two statements referred to above, viz. : — " On the new plan we may label a bit of energy and trace its motion and change of form, just as we may ticket a piece of matter so as to identify it in other places under other conditions ; and the route of the energy may be discussed with the same certainty that its existence was continuous as would be felt in discussing the * It should be noted that according to these two conceptions the third law, though expressed in the same words, i.-i not the same law. In the one case it applies to all material bodies, whether in contact or at a distance. In the other it applies to all bodies in contact, whether they are material bodies or elements of the medium. t It should be noted, however, that the law of conservation during transference which, notwithstanding the reiteration of his old claim, is all that Prof. Lodge now considers himself to have deduced, does not of itself", as he seems to suppose, justify the assertion of the above proposi- tion. For it is consistent with this law that energy may, as he says, " leak away in some silent unobtrusive fashion." If it may thus leak out of observation, it may also leak into observation. Since, then, some of the energy which appears within a bounded region may have got there by this process of unobtrusive leakage, and since the above law can tell us nothing as to how it got there, this law cannot of itself justify the above proposition. It cannot therefore even in this first sensf be called a law of identitv. A 258 Prof. J. G. MacGregor on the roved. We cannot be said to be «. leroy in its wanderings unless we route of ome lost luggage which has turned up at a distant station ii however battered and transformed a condition " *. I need not discuss here the question whether even matter can be said to have the kind of identity here specified, whether in fact we can label a bit of it and follow it in its wanderings. We certainly cannot, in general, do so practically. Whether or not we can do so ideally may, I think, be found to depend upon our hypothesis as to the constitution of material bodies. That energy, however, has this kind of identity seems to me to be a much more definite proposition than the one considered above, and not by any means to be implied in it. Indeed, that the contact-action conception of energy does not confer upon it the capability of being +hus followed in its motion may, 1 think, be able to follow a " bit " of are able at all stages to localize it. If in the course of its peregrinations it enter a system of bodies and get so hidden away that we can only say of it that it is in some body or other of the system or is distributed in unknown proportions among them, then we have lost it even more completely than we should have lost our luggage in a railway collision if we knew onlv that it was distributed somewhere among the debris. Now we have seen that the assumption of contact- action alone does not enable us to localize potential energy. While, therefore, the introtluction of the axiom of contact- action confers upon energy a certain kind of continuity, telling us that it must pass from one body to another or to others, it does not enable us to follow a bit of energy and to trace its route, because it does not in all cases enable us to localize it. If we wish to be able to trace its route com- pletely, we must introduce a further axiom which, with contact -action and the third law, will make our conception of the transference of energy sufficiently complete. Then we shall be able to localize energy under all circumstances, and the first condition of following its motion will be satisfied. But more than mere localization is necessary in order to label and follow a " bit *' of energy. We nuist also be able to distinguish it from other bits when several of them at the same time get into the same body ; and here the chief diffi- culty seems to arise. It is easy enough to frame an hypothesis of acting mechanism which will localize energy in all cases, provided we do not mind much whether or not it coordinates for us dynamical phenomena generally ; but how we are to distinguish between the portions of energy which are trans- ferred simultaneously, say, to a particle by the various elements of the medium in contact with it, is not so apparent. To my * Phil. Mag. 7)] vol. xix. p. tH2. Hypotheses of Dynamics. 259 mind they must get mixed. Yet, before we can bo said to follow a bit of energy as wo do a labelleil portmanteau, we must either have some means of making this distinction, or we nmst frame our hypothesis of acting mechanism in such a way us to exclude the possibility of two bits of energy getting into the same body at the same time. Prof. Lodge refers to I'rof. Poynting's ])aper on the " Transfer of Energy in the Eleetromagne'ac Field"* as an illustration of the [)ovver which the contact-action hypothesis fives us of labelling and following bits of energy. Prof. *oynting*s results, however, were based not only on the assumption of contact-action, but on other hypotheses as well, which made his ariom of acting mechanism, if not complete, at any rate nmch more complete than contact-action alone would have made it ; and, moreover, if 1 understand him aright, he did not profess to label and follow the bits of energy distributed in the fields which he investigated. (7) The complete Transformation of Energy during Transference. in the present paper Prof. Lodge gives a formal demon- stration, and a discussion of illustrative instances, of bis pro- position that, according to the contact-action conception, "energy cannot be transferred without being transformed "f. I need not enter into a detailed criticism of the demonstration. It is sufficient for my purpose to draw attention to two points: — (1) The demonstration itself admits the possibihty that a body may act as "a mere transmitter, not itself active, only passing on what it receives." and applies to bodies not acting in this way. But a body cannot pass on the (inorgy it receives with- out the energy being first transferred to it and subsequently transferred by it. The d(;monstration, therefore, admits that energy may in certain circumstances be transferred Avithout being transformed, excludes such cases from consideration, and restricts itself to other r-asesj. (2) The demonstration is entirely qualitative. It is shown that in these other cases of action between two bodies, if one body lose, or gain, kinetic * Phil. Traua. 1884, pt. ii. p. 343. t Phil. Mag. [o] vol. xix. p. 480. The formal .statement of the pro- position in the present paper is much less precise than in the paper just cited. He says, '' My proposition was that the change of form is always from kinetic to potential or vice versa,'" though he certainly would not have set himself to prove anything so obvious. The context, quoted below, shows that it is the complete transformation of energy during transference that the demonstration is held to prove. X It should be noted, however, that on p. 33 Prof. Lodge speaks of the treatment of potential energy as be'ing " conveyed elsewhere as a simple flux without transfer or transformation," as '' blindfold treatment " which " doeei not exhaust the matter.' 1 KB 260 Prof. J. G. MacGregur on the energy, or potential energy, the other body will gain, or lose, potential, or kinetic, respectively, /". e., that in these cases there will be some transformation. In order to prove com- plete transformation, it would be necessary to show that the potential or kinetic energy lost or gained by the one body was equal to the kinetic or potential gained or lost respectively by the other. Thus even if wo admit the validity of the argument without criticism, all that it proves is that, except in certain specified cases in which transference occurs with- out transformation, transference always involves some trans- formation, which is equivalent to the affirmative part of my conclusion that "transference of energy will in general involve partial but not complete transformation." Besides this demonstration Prof. Lodge gives a discussion of two examples. The first, the loaded air-gun with its muzzle plugged, is an example of the transformation of potential energy during transference ; the second, the impact of a couple of equal elastic rods moving end-on, exemplifies the transformation of kinetic energy. With regard to the former, Prof, Lodge says : — " Tha compressed air has jiotential energy; on its release its energy is transferred to the moving wad, which instantaneously hands it on to the air near the muzzle, compressing it, and thus retransforming itself into the potential form.'' This seems to me a very inadequate account of what occurs. It assumes the wad to be a rigid body. As it is not, however, the first effect of the expansion of the compressed air must be to compress the adjaceat end of the wad. But compression involves the relative motion of its parts. Hence during the first small expansion of the compressed air the wad mu*^^ simultaneously gain both potential and kinetic energy ; and therefore the potential energy lost by the compressed air has not been completely transformed in transference. If the wad be " dematerialized " and the example treated from the point of view of the kinetic theory of gases, the energy of the com- pressed air is practically wholly kinetic ; and if, as Prof. Lodge says, the collision of two particles is m(»st conveniently thought of in terms of elasticity, the question which has to be settled is what transformation occurs during the collision of two elastic particles. Treated in this way, the example thus resolves itself into the second example * — the impact of two elastic rods. * I need not make further reference to Dr. Lodge's discussion of thid example from the point of view of the kinetic theory of gases, partly for the reason mentioned and partly because a portion of it is said to be " true in one sense, but not a titial or complete statement." Uypotheseis of Dtjnamics. 2G1 1 need not enter into a criticism of the discussion of this example, the conclusion bein^ quite sufficient for my purpose. '• By the considt-nition of instances," Prof. Lodge says, " we have thus been led to the induction that energy can he trans- niitt(^d without obviouc cliango of form by substances with infinite properties, e. g. by an incomj)ressible solid ; all mole- cular processes being eitlier non-existent or being ignored ; but that with ordinary matter there is always some percentage of obvious transformation, though we may apparently have all grades of it from complete to very small." This is of course partial, but not comj)lete transformation. "Thiidving of these impact cases alone, it might appear as if I had been overhasty in saying that the whole of energy must be transformed when it is transferred*. Yet observe that it has to pass through the intermediate condition. A row of ivory balls in contact has another thrown against one end, and from the other end one leaps off. The energy has been transmitted through the row somewhat as it is transmitted througli the compressed strata of two impinging rods. Yet if the elastic connexions of every stratum are attended to, and if these be regarded as massless, 1 think it will be found that all the transmitted kinetic has really passed through a momentary existence as potential." Thus, in order to uphold the com- plete transformation of kinetic energy during transference, it is found necessary to assume the rods to consist of strata alternately massive and massless. Similarly, in order to make good the complete transformation of potential e ""ergy it would be necessary (as in the former example) to assume the rods to consist of strata alternately rigid and elastic. " The fact of necessary transformation," he continues, " is not so obvious when you come to look into some of these special cases ; bui I would refer once more to the proof given at the beginning of Part IV., which seoms to me conclusive as to essential fact." Prof. Lodge therefore finds it difficult to make the accuracy of his thesis obvious in such exam])les, and falls back upon the general demonstration, which, as shown above, even if it be admitted without criticism, proves only partial trans- formation. " The difficulty arises because when an elastic body is struck (say a massive molecule with a massless spiral spring connexion) it begins to move a little directly the s^^Jng is the least compressed, and is moving half speed w^hen the spring is fully compressed." In my paper (p. 140) I pointed out this " difficu'ty " as standing in the way of the doctrine * See footnote f to p. 259. 202 I'rof. .1, a. MiiC'(T!iVi{or nn the of complete transforiimtion, in tlie followini^ words : — " If one body exert on another a certain force tlirounjh a certain dis- tance, the same work is done on it whether the former body lose kinetic or potential energy in doing the work ; while the eflFect produced in the Latter body will in general be a change both in its motion and its state of strain, i.e., both in its kinetic and its potential energy. Thus, whether the former lose kinetic or potential energy, the latter will in general gain both, or transference of energy will, in general, involve partial but not complete transformation." Prof. Lodge thinks, how- ever, the ditficulty may be met. For he observes in con- clusion : — " But 1 venture to say that on anj^ view of the identity of energy the bit of kinetic which it first attains is a bit of energy tliat has boon transmitt(;d through the elastic stress of the spring, and that just as the second half of the enei'gy must admittedly exist in the spring before it can reach the mass, so the first half has already passed through the spring and has reached the mass only after transmutation, although the transformation is disguised while the trans- ference is obvious." The difficulty therefore is overcome only by assuming bodies to consist of massive molecules connected by a massless but elastic medium. Compare with this the following from my paper (p. 141) : — " If we assume the particles [of bodies] to be rigid they can of course have kinetic energy only. If the medium be assumed to have no inertia, its elements can have potential energy only. Hence if both assumptions bo made, transference of energy between the particles and the medium must involve complete transforma- tion, while transference from element to element of the medium must occur without transformation. If, however, both the particles and the medium be assumed to have both inertia and elasticity, the transference of energy will, in general, in- volve only partial transformation, whether it occur between the elements of the medium or between the particles and the medium." It will thus be obvious that the necessity, which Prof. Lodge here acknowledges, of making definite assumptions as to acting mechanism, when he endeavours to follow his contact- action energy and to make its transformations agree with those prescribed by his thesis, is in entire agreement with what I have pointed out above, viz., that the assumption of contact- action alone is not sufficient for the purpose of completely localizing energy. Without being conscious of the fact, he assumes, not only contact-action, but also that bodies consist of massive and rigid particles, and that the medium through IJi/jwt/it-.v's iif Ihjnamics. 2r>.'» which thoy act on one another is nuissU^ss and elastic*. 13iit even with this axiom of actint]; incchanistn his thesis as to coniploto transformation (Uiriii^j; Iransl'erence liolds only for transferences between particles and elements of the medium, and not for transferences from element to element of the medium. (8) The Complete Iramferenee of Energy during Tranaformation. Besides the proposition just con««idored, Prof. Lod^e, in the paper referred to above, asserted also that " energy cannot be transformed without being transferred." Although I called this proposition in question also, no reference is made to it in the present paper beyond a reiteration of the assertion (p. 16). In a synopsis of the paper, however, published by Prof, Lodge himself t, the admission is made that the assertion is incapable of proof. For we tind it laid down, as the fifth axiom of dynamics, that " energy which is not being actively trans- ferred from one body to anofclier remains unaltered in quantity and form." Prof. Lodge's right to enunciate this proposition as an axiom may be judged by his own standard. According to him (p. 3 of his paper), in setting forth an axiom, (1) regard must be had to the ex[)erience of the human race, (2) hundreds of instances should be adduced in whicli it holds, (3) a few special cases should be critically examined and in no case found to fail, and (4) '' contrary instances " should be called for. Most of these regulations, which are obviously intended to prevent people from carelessly and thoughtlessly enunciat- ing axioms, are admirable. With the exception, perhaps, of the appeal to the experience of the race, such formulators of axioms as Galilei, Newton, and D'Alembert followed them, especially as to the critical examination of special cases. But Prof. Lodge, in the present instance, ignores them all. He does not show how the proposition in question appeals to the race ; he gives no instances in which it holds ; he examines no special cases critically ; and he makes no reference to a contrary instance which I ventured to bring forward in my paper (p. 141). Wo are forced, therefore, to conclude that according to his own regulations he is not yet in a position to enunciate this proposition as an axiom. * On p. 21 he speaks of " the potential energy of the particles of a opring," thus assuming its particles to be eluFtic, but this occurs in a paragraph which is " true in one seni>3 but not a final or complete statement." t ' Nature,' vol. xlviii. p. 62. iS 264 On the Uijpothe«e.H of Dynamics. But there is another ground on which this course is seen to bo for him entirely illogical. For though his having taken it is an admission that ho regards the proposition in question as incapable of deduction, he has been hasty in reaching this conclusion. We have seen that, in order to maintain complete transformation during tranfe^ence, he has confessedly to assume that bodies consist of massive particles without elas- ticity, and that the medium is elastic but without inertia. Now, as pointed out in my paper (p. 141) : — "That energy cannot be transformed without being transferred must of course be true if bodies consist of particles with inertia but without elasticity, and if tho medium co'inecting them possess elasticity but not inertia." This is surely quite obvious. Hence with the assumptions already made by Prof. Lodge the proposition under consideration is capable of deduction. For him therefore it cannot be an axiom. For those of us who do not hold to his theory of the con- stitution of bodies and media, the proj)Osition is of course not capable^ of deduction. Whether or not we are to regard it as axiomatic must depend on whether or not it may be shown to be capable of coordinating dynamical phenomena generally. Edinburgh, July 2l8t, 1893. k i w