^^a ^^^. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 g m 12.2 itt Z Itt IgO m ■ * r CIHM/ICMH Microfiche Series. dlHIVI/ICMH Collection de microfiches. ' Camdian Institut* for Historical IMicroroproductiona / inatitut Canadian da microraproduct!ont hiatoriquaa ■ * Taehnioal and Bibliographic Notaa/Notaa tachniquas at bibliographlquaa Tha Inttituta haa attamptad to obtain tha boat original copy availabia for filming. Faaturaa of this copy which may ba bibliographically uniqua, which may altar any of tha Imagaa in tha raproduction. or which may aigniflcantly changa tha uauai mathod of filming, ara chaclcad balow. D D D D Colourad covara/ Couvartura da coulaur |~~| Covara damagad/ Couvartura andommagAa Covara raatorad and/or laminatad/ Couvartura raatauria at/ou paiiiculAa Covar titia miaaing/ La titra da couvartura manqua Colourad mapa/ Cartaa gtegraphiquas an coulaur Colourad ink (i.a. cthar than biua or blacic)/ Encra da coulaur (i.a. autra qua blaua ou noira) I I Coloured plataa and/or illuatrat!jna/ Planchaa at/ou illuatrationa an coulaur QBount Raiii Bound ¥vith othar matarial/ avac d'autraa documanta Tight binding may cauaa ahadowa or diatortion along Intarior margin/ Liiroliura aarr*a paut cauaar da I'ombra ou da la diatortion la long da la marga IntAriaura BlanIc laavaa addad during raatoration may appaar within tha taxt. Whanavar poaaibia, thaaa hava baan omlttad from filming/ II aa paut qua cartainaa pagaa blanchaa ajoutAoa lora d'una raatauratlon apparalaaant dant la taxta, mala, ioraqua cala 4tait poaaibia, caa pagaa n'ont paa 4t4 fllmdoa. Additional eommanta:/ Commantairaa auppMmantairaa: L'lnathut a mlcrofilmA la malHaur axamplaira qu'il liil a 4t* poaaibia da aa procurar. Laa d4taila da cat axamplaira qui aont paut-#tra uniquaa du point da vua bibliographiqua, qui pauvant modifiar una imaga raprodulta, ou qui pauvant axigar una modification dana la m*thoda normaia da filmaga aont indiquta ci-daaaoua. Th to D D D D D n D D Colourad pagva/ Pagaa da coulaur Pagaa damagad/ Pagaa andommagtea Pagaa raatorad and/or laminatad/ Pagaa raataurtea at/ou paiiieulAaa Pagaa diacoiourad, atainad or foxad/ Pagaa dAcolortaa, tachatAaa ou piquAaa Pagaa datachad/ Pagaa oAtachAaa Showthrough/ Tranaparanca Quality of print variaa/ Qualit* inigala da I'impraaaion Includaa aupplamantary matarial/ Comprand du matArial auppMmantaira Only adition availabia/ Saula Adition diaponibia Pagaa wholly or partially obacurad by arrata allpa. tiaauaa, ate., hava baan rafilmad to anaura tha baat poaaibia imaga/ Laa pagaa totalamant ou partiallamant obacurdaa par un faulNat d'arrata, una palura, ate, ont AtA filmAaa A nouvaau da fapon A obtanir la malllaura imaga poaaibia. Th po of fill Orl b«l thi aio oti fin aio orl ahi TIN diff rigl Thia itam la fllmad at tha raductlon ratio ehaekad balow/ Ca documant aat f llmA au taux da rAduetton inittquA ci-daaaoua. 10X 14X ItX 22X 2SK aox ] 12X IfX SX MX 32X mmm_ The eopy fllm«d hmn Hm b««n raproduoad thankt to th« oMMrotitv of: UniMnHy of AllMrta Edmonton L'oxomploirt f ilm4 f ut roproduit grico A la gAnArooitA da: Univtnity of Albarta Edmonton quality laglbllity Tha imagaa appfta^ing hara ara tiia poaalbia eonaMaflnfl tha oondition of tha origifiiil oapy mnd In kaaplng fflniing contvaot apaalfioatlOfia< Original eopiaa in printad papar eovara ata fHmad baglnning with th^i front oovar and anding on tha laat paga with a printad or IHuatratad impraa> tlon. or tha baok eovar whan approprlata. AN othar ariglnal eapiaa ara fHmcd baglnning on tha firat paga with a printaJ or iihiatratad Impraa- •ion. and anding on tha iaat paga with a printad or iSluatratad impraaaian. l.ao imagaa tuivantaa ont AtA raproduitas avac la plua grand aoln, eomptc taiiu da la condition at da la nattat* da raxamplairr film*, at •» conformity avac laa conditions du contrat do fllmaga. Laa axampiairaa originaux dont la couvartura ^n papiar aat ImprimAa tont flimta on commandant par la pramiar plat at an tarminant toit par la damMra paga qui comporta una omprainta d'Impraaaion ou d'lllustration. soit par la tacond Piat. salon la eaa. Toua laa autras axampiairaa originaux aont flimte an commandant par la pramMra paga qui comporta una amprainta dimpraaslon ou d'iHustration at an isrminant par la damlAra paga qui comporta una talia amprainta. Vha laa^ raeordad frama on aaeh nd cr of lch o shall contain tho aymbol -n^^ (moaning "CON- TINUED"), or tho aymbol ▼ (moaning "END"), Un daa symbolas suivanta apparattra sur !a damiira imaga da chaqua microflcha. salon la caa: la symbola ^^ signifia "A SUIVRE". la symbols ▼ signifia "FIN". IMapa. plataa. chMta. ata.. may ba fNmad at d i f fa rant niduetion ratloa. Thoaa too larga to ba antlraly ineludad in ono axpoaura ara fHmad baglnning in tha uppar iaft hand comar. loft to right and top to bottom, aa many framaa as raquirad. Tha following diagrama illuatrata tha Las cartas, planchas. tablaaux. ate, pauvant Atr fiimAa A daa taux da rAduction diff Arants. Ixraqua la doeumant aat trop grand pour Atra raproduit an un saui clichA. il aat fiimA A partir da i'angia supAriaur gaucha. da gaucha A droita. at da Itaut 9n bmt. an pranant la nombra d'imagaa nAcaaaaira. Laa diagrammaa suivantt iiluatrant la mAthoda. 1 2 3 1 2 3 4 5 6 ■ ill III I n BcpHntedfrom the Mokphlt Not.o., or t„h Rova. A„«OKOM,OAt Soanv. Vol, Lll. So. c. .1:1 ON THE DYNAMICS OF THE EARTH'S ROTATION. WITH RESPECT TO THE PERIODIC VARIATIONS OF LATITUDE. BT SIMON NEWCOMB. ii 336 Trof. Neweomb, On the Dynanici t" 5i ! I i i i On the Dynamits of the EartKa Botatton^ with respect to the Periodie Variations of Latitude. By Simon Neweomb. The recent remarkable discovery of Mr. S. C. Chandler, that the axis of rotation of the Earth revolves aroaud the axis of mazimnm moment of inertia in a period of aboat 427 days, is worthy of special attention.* At first sight it seems in complete contradiction to the principles of dynamics, which show that the ratio of the time of such a rotation to that of the Earth's revolu- tion should be equal to the ratio of the polar moment of inertia of the Earth to the difference between the equatorial and the polar moments. Representing these moments by A and G, it is well known that the theory of rotation of a rigid body gives the equation T being the period of rotation of the pole in sidereal days. Now the ratio in question is given with an error not exceed- ing a few hundredths of its total amount by the magnitude of the precession and nutation. The value found by Oppolzer is -L , giving the time of rotation as 305 davs. 305 This resnlt has long been known, and several attempts have been made to determine the distance between the two axes, especially at Poulkova and Washington. A series of observa- tions was made with the Washington Prime Vertical Transit during the years 1862-1867, molading six complete periods of the inequality. Thus the determination of the coefficient and Eero of the argument is completely independent of all sources of error having an annual or diurnal period. Such errors are * Attronomical Journal, Numbers 348, 249. March 1892. of ihe Earth's Rotation ete. 337 liable to affeot the determinatioa aniess it is oontinaed over this period. A preliminary disoussion of the observations, which was made at the request of Sir William Thomson, and poblished by him, gave a coetfioient of o"'05 for the Inequality. A more com- plete disoDssion, undertaken qnitu recently, rednces the co- efficient to o"'03, corresponding to a distance of three feet between the two axes. This result was quite within the limits of errors of observation, and seemed to show that there was no appreciable difference between the two axes. This result was in complete accordance with the conclusions reached from the Poulkova observations, and seemed to show, beyond doubt, that there could be no inequality of the kind looked for. Mr. Chandler's discovery gives rise to the question whether there can be any defect in the theory which assigns 306 days as the time of rotation. The object of this paper is to point out thai there is such a defect — namely, the failure to take account of the elasticity of the Earth itself, and of the mobility of the ocean. The mathematical theory of the rotation of a solid body, on which the conclusions hitherto received have been based, pre- supposes that the body is absolutely rigid. As the Earth and ocean combined are not absolutely rigid, we have to inquire whether theii- flexibility appreciably affects the conclusions. That it does can be shown very simply from the following consideration : — Imagine the Earth to be a homogeneous spheroid, entirely covered by an ocean of the same density with itself. It is then evident that, if the whole mass be set in uniform rotation around any axis whatever, the ocean will assume the form of an oblate ellipsoid of revolution, whose smaller axis coincides with that of rotation. Hence, the axes of rotation and of figure will be in perfect coincidence under all circumstances. To apply a similar reasoning to the case of the Earth, imagine that the axis of rotation is displaced by o"'20 from that of greatest moment of inertia, which I shall call the axis of figure. Then, with an ocean of the same density as the Earth, its equator would be displaced by the same amount. The ocean level would change in middle latitudes by about one inch at the maximum. But this change would have for its effect a corresponding change in the axis of figure. As the ocean covers only three-fourths of the Earth, the axis would be displaced by three-fourths of the distance between the two axes, n«re ocean and Earth of equal density. But, as the density of the Earth is some five times as great, the actual change would be only one-fifth of this. It would even be less than one-fifth, because the displacement of the ocean equator woald be resisted by the attraction of the Earth itself. The exact amount of this resistance cannot be accarately given, but I think the displacement would thereby be 338 Tfof. Neweomb, On the Dynamie$ LII. 5, t I I redaoed to one-half. I therefore think that one«fonrteenth would he an approximate estimate of the dinplaoement of the axia of fiffore, in consequence of the movement of the ocean. As Mr. Chandler's period requires a displacement of two-sevenths, the ocean displacement only accounts for one^fuurth of the difference. The remainder is to be attributed to the elasticity of the Barth itself. It is evident that the flexure caused by the non- coincidence of the two axes tends to distort the Earth into a spheroid of the same form as that which the ocean assumes, and thus to bring the two axes together. We have now to show how this deformation of the Earth changes the time of revolution. Let us imagine ourselves to be looking down upon the North Pole, and let P be the actual mean pole of the Earth when the two axes are in coincidence, and B the end of the axis of rotation. Then, in consequence of the rota- tion around B, the actual pole will be displaced to a certain point, P'. Now, the law of rotation of B is such that it constantly moves around the instantaneous position of P' in a period of 305 days, irrespective of the instantaneous motion of P' itself. In other words, the angular motion of B at each moment is that which it would have if P' had remained at rest. Hence, the angular motion as seen from P is less than that from P', in the ratio of F B : P B. But, as B rotates, P' continually changes its position and rotates also, remaining on the straight line PR Thus the time of revolution of B around P is increased in the same ratio. We may next inquire what degree of rigidity the Earth must have in order that the total displacement of the axis of figure produced by the change in the centrifugal force may be two- sevenths that of the displacement of the axis of rotation; in other words, that the ratio P'B : PB may be 5 : 7. A rigorous treatment of the problem is scarcely possible, as the rigidity probably varies from the surface inward; I shall therefore only attempt a rough estimate, founded on certain conclusions as to the deformation of a rotating spheroid reached in Thomson and Tait's Natural Fhilosuphy. To proceed in the simplest way, I shall assume the earth to have the rigidity of steel, and inquire to what displacement the axis of figure would be Hubjeot, in consequence of the centrifugal force arising from a rotation around an axis differing from the normal axis of figure. Conceive a solid sphere, of the same size and general con- stitution as the Earth, to be set in rotation like the Earth. Let c' be the ellipticity induced in it by the rotation, and let < be the actual ellipticity of the« £.irth. We shall then have a superposition of two ellipticities, the one c, such that P is the pole of figure ; the other, c' such that B is the pole of figure. P' being the pole arising from the combined ellipticities, I assume that we have the proportion PP': •'=?'»:€. To find tljie value of c' I start from the conolusion of Thomson 1 March 1891. of the Earth's Botation etc. J39 and Tait (§837), that a ball of steel of any radina rotatitif? with an equatorial velocity of 10,000 centimetres per second will be flattened to an eUipticity of . The Earth'a equatorial velocity is 4*65 times this. Its density is less than that of steel : the density which we should assume is not the actual mean density but a mean in which greater weight is given to the superficial portions, because these have the greatest centri- fugal force. Probably the actual mean to be adopted is 06 of the density of steel. We have, therefore, neglecting the effect of gravitation, ,' „o6 x465V I " 7"o 557" But the deformation of the Earth is resisted by the gravitation of its parts. By a theorem given by Thomson and Tait, we shonld have, takmg this efiect into account — Hence we have + -557 + 292 = 849. iBon Hence, considering only the solid Earth, PP':P'Il-292:849. We have already concluded that the motion of the ocean will shift P' oue-l'ourteeuth of the way from P' to B. Hence, liually, PP' :P'R= 353:788 PR :P'R = 1142:788. Time of revolntion of pole » 443 days Period for a rigid earth = 306 „ Compntnd increaite of perioda 137 „ Observed increase of period «> 121 „ The conclusion is that the Earth yields slightly less to the centrifugal force than it would if it had the rigidity of steel, and that it is consequently slightly more rigid than steel. We have next to consider the effect of viscosity of the earth. Those geologists who have given special attention to the subject regard it as well established that the Earth yields under the weight of deposits as if it were a thin orast floating apon a liquid interior, and must therefore be a viscous solid, if a solid at all. The effect of viscosity is that the normal pole P of the Earth would be in slow but continfaous motion towards the revolving pole B. Both P and B would then describe logarith- mic spirals, so related that the tangent to the inner spiral at the position of P at any moment wonld pass throngh the position of R at that moment, and out the B spiral normally. Thus the line PB would diminish from century to century by equal 340 Vrof. Neweomb, On the Dynamiea tii. 5, I i I frnotions of its amount in eqnal times. Thus the poles would eventually appear to meet, unless separated from time to time by the action of causes changing one or both of them. Since the position of the pole of figure of the Earth may be supposed to nave been originally determined bv the rotation itHolf, and continually to approach the pole of rotation if it were very slightly separated from it, the presumption would appear to be that the two poles would now be in apparent ouiucidence, in the absence of disturbing causes. Moreover, the evidence of the most accurate observations hitherto made with Prime Vertical Transits seems to show that the separation of the two poles at the epochs 1842 and 1864 could scarcely have exceeded the tenth of a second. But observations made with probably equal exactness at the present time seem to show, according to Mr. Chandler, a separation of o"'3. It would seem, therefore, accepting these provisional numerical results, that some disturbing cause has acted. A vera catua was pointed out some years ago by .Sir William Thomson, in the motions of the winds and oceans, and especially in changes in the polar ice- cap, In order to have its greatest efiuot such a movement of matter must occur in the middle latitudes ; a change in the polar ice-cap would be the less appreciable in its effect the nearer i: occurred to the pole. A heavy snow-fall over the whole of Northern Asia, unaccompanied by a corresponding fall on the American continent, would undoubtedly cause a slight displace- ment ; but I doubt whether the greatest effect of this kind oould amount to o"'05. But we have also to consider the effect of an annually repeated disturbance of this kind. Mr. Chandler's period is such that the pole of rotation makes six revolutions in seven years. Hence, during one-half the period of seven years, the effect of an annually repeated cause will be cumulative. In a recent volume of the Bulletin Attronomiqiie, Mr. Badau has investigated the effect of an annual periodic change in the position of the Earth's axis of figure, and shown that it will be multi- plied three times, in consequence of this oumulative effect. But bis analysis rests on the hypothesis of a 306-day period. It is worth while to show how such an annual cause would act when we adopt Mr. Chandler's period. Maroh 1899. of the Earth's Rotallon ete. 341 Let Q bo the moan poaition of tho pole of ^anre of the Enrth, and let uh aHNamo that tho actual |h>Iu P rovofves around it in a radius a, and iu a period of one year. Lot R be the position of tho pole of rotation at any time. Then, at each moment, R ia rovolvinff around the fixed poflition P with a uniform motion, which, if continued, would cause it to complete a revolution in 437 days. Let as put n, tha msan motion of the radius PQ ; ft, the mean motion of R around the ponition of P ; X, y, the rectangular oo-ordinateN of R refurrud to 4 aa an origin. The law of rotation then givoH tho equations dx dt ' — 1*1/ + an Bin (»< + c) f •• fuc - an cot (nt + c). dt The integration of these equations gives V -> a sin M< + /i cos /i/ - ^^ sin (»< + 0), a and P being arbitrary constants. Substituting for fi and n their nnmerical '"lues, we have, approximately, X a a cos M^ - /S sin ^< -I- 6a cos (»t -f e) y-asin^<4jSco8M<-<-6<'Bin {nt + c). Such a rotation as we have supposed, around a circle of o"'05 in radius, would suffice to produce anomalies as large as those actually observed. If the winters in Siberia and in North America occurred at opposite seasons, we should have no difficulty in accepting the sufficiency of annual falls of snow to account for the anomaly. But, under the actual circumstances, we must await the results of further investigations into the whole subject. SpottUttooie 4 Co. Printers, Xtw-tlrttI iSfiiarr, lotidim.