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No. 4..-V ON THE SOLUTION OP A CERTAIN DIFFERENTIAL EQUATION WHICH PRESENTS ITSELF IN LAPLACE'S KINETIC THEORY OF TIDES. - '• - By Mr. Geoboe Hebbert Lino, New York, N. Y. -■-•<■■• > [Siibmittad in pfxrtinl fulfilment of tbe requirements for .he degree of Doctor of Pbilosopliy iu tlic Faculty of Pure Science of Columbift UniverBity iu the City of New York.] ,:^i.,\ ■ 'X '■' !j ; „. X ■<1,., ■. '■■'■' '% r^^ m ^»^ *■*.-<*,•,*,, c rp.'.. \/.sI ^^f ON THE SOLUTION OF .\ CEIITATN DIFFERENTIAL EQUATION WHKJH PRESENTS ITSELF IN LAPLACE'S KINETIC THEORY OF TIDES. By Vu. Gkoiuik Mkubeiit IjINn>,' a parallel Tide nt point distant .) from the boundary of canal, .... Canal of negligible width, VIII. Summary of IIesiilts. Suiniunrv of I IV, Suinnuiry of V VII, 90 1)7 !)7 !)0 99 101 101 102 103 104 107 109 110 111 113 114 11.-. no 117 119 120 122 123 124 124 96 use. ON THE HOLUTION Ol' A CEUrAIN DIFFEllESTIAI, KtM'ATION I. Intuoductory. 1. Ohjiu-tn iif till' /)'ij>er. In his discuHsion of the kinetic theory of tides, Liii>laco found that tlu! function expressiu}j; the hei^Iit of the tide at ii '.Mven point due to tiie attraction of tlie disturl)iu},' hody satistied a certain ditfereu- tial eiiuation. l-'indin},' liiniself unable to ohtain tlie ^jfeneral soUition of the dilferential equation, lie applied himself to the discussion of several partimihir eases which aris(! when certain assumptions are made re<,'ardinj,,' the physical constitution of the ocean. Oni; of the cases lus treated was that of tlu! semi- diurnal tide when tlu^ dei)th of the ocean is supposed to be constant. In the course of his treatment of this case certain considerations enter which have given rise to much discussion. It is pro[)osed to devote some attention to this ease, and it is hoi)ed to extend the treatmtJiit of this case so as to include some phases of it not previously treated. While it is generally conceded that the facts in r(!j^ard to the disputed point referred to, have been made evident, yet the methods of i)lacing those facts in evidence have been called into question by several writers on the subject, and do not appear to be the most satisfao tory ones that are available. II. HisTOKicAL Sketch. 2. Orii/i)) of the pruhleni. As just mentioned, the subject to be discussed was first treated by Laplace. His kinetic theory of tides is set forth in the Mecaniqne Celeste, and the part with which we are concerned is to be found in Livre IV of that work, his solution of the differential ecpiation being given in Article 10. Considerable time had elapsed between his Hrst di.scussion of the subject and the publication of his great work. The earliest presentation of his treatment of the subject was contained in a memoir* presented to the Academie des Sciences, and contained in Tome IX of the (Euvres de Laplace. He has sought a solution of the ecjuation in the form of a series of positive entire powers, and has made use of a certain infinite continued fraction in the evaluatijii of one of the coefHcients of the series. The correctness of the value found by his method has been questioned. As the solution in the series form was made the basis of his calculations, it was of great importance that no mis- take should be made in the determiuat: m of the coelHcieuts, and more esjie- cially in the determination of those occurring early iu the series. * Keiheiches Kiir phisieurs poiuts du Systeme du Mouae. Meraoire- }'Tt. fu his oiirly -lUMiioir liapliice luiH f,'oiio sonuswlmt nioro into detail, ami tlio mothod by wliicli lio dctciiiiiiKHl tlm value of the eoisrtitiiisiit is eleavly shown. In tiio hiter work he has omitted a Rieat part of the exphmation, and has contented himself with exjiressin^' the quantity in tlie form of Uie eoiitinued fnietion to whieli reference has heen made. The later presentation of the sul>jeet has heen the more aeeessiMe of tlie two, and on it all the later writers appear to have based their remarks eon- cerninf; [.aplace's method, wliile the orij^inal ])resentation has been overlooked. Attentio'i has been called to it by Prof. Lamb in his recent work on Hydro- dynamics, and to him seems to be due the it'discovery, so to speak, of the memoir. Laplace's evaluation of the coetHcient was objected to bs Sir G. 13. Airy,* and later the same objection was made by Mr. William Kerrel.t A defence of Lajdace w;ts made by Lord K<^lvin in the rhilosopliical Ma^'a/.ine for September, 1875. The October number of the same journal for 1875 con- tains a note writttiii by Airy in which ho reatHrms his objections to Lajjlace's result, and appears not to re<,'ard Kelvin's reasoninj,' as convincing?. The num- ber of this journal for March, 1>S7(), contains a reply by Ferrel to the arguments of Lord Kelvin. Prof. G. H. Darwin in the Encyclopedia Britannica:|: gives in more detail Lord Kelvin's argument. His treatment of the subject may also bo found in IJasset's Hydrodynamics, Vol. II, and Basset has briefly referred to the subject in a foot note.jj The latest contributions to the subject are believed to be the two papers by Ferrel which appear in Volumes 9 and 10 of Gould's Astronomical Journal. The latter of the two papers may also be found in the collection of papers || on the " Mechanics of the Atmosphere " edited by Prof. Cleveland Abbe, lleference may also be made to Professor Lamb's Hydrodynamics, in which attention is called to Laplace's original memoir. 4. S)//ii)/iniK iif pri'i'hivti niiitrihiitidiin. Before treating the problem analytically it will be useful to sketch the arguments of Laplace and those who afterwards treated the subject. Laplace, assuming that the solution of the e([uation could be ex])ressed by means of a Taylor's series, substituted such a series with undetermined coefHcients in the ditierential equation, and was able to determint; all the coefficients of the series in terms of one of them, which remained arbitrary. He had previously argued that it was not necessary to obtain the general solution of the ecpration, since, as he atHrmed, the arbitrary coustsiuts would be tletermined by the initial comlitions of the water and would introduce ettects dependent on this initial condition, which etfe>'ts ought to be ♦ Article, " Tides uud Waves," Eneyoloi. lia Metropolitaiui. + " Tidal Uesearthes," Api>eudix to ITni.ed ;^tutes Ciiast aud Geodetic Survey Ki'ijort for IH',4. X Article, " Tides." Eucyolopedia Britauuica. § Basset, Vol. II. 1>. 218. II No. 84a, Siuithsouiaii MisctUaueous Coutribiitions. w ^ m n m ^w ^ 98 LINO. ON THE SOLUTION OF A CEUIAIN DIFFEUENTIAL EQUATION (lisregavdod, Hiiico iu tlie case of the sea they would Vm^ ago liavo l)i)i'ii over- come by friction. Considoriug, then, that any particuhir integral wus HutH- d<>nt, h(- proi'oedod to choose the most Hatisfaotory value of the eoerticiciit. Mis metliod of deciding tiie proper value of the coetHcient will lu' given in Section III. It enabled him to satisfy himself that a comparatively few terms of his series would give the result with a very small error. From the form in which th(! result is set forth in tint M«'cani(pie Celeste it apjjears, however, that he made the assumption that the ratio of any coi>thcient in his series to the preceding one becomes ultimately smaller than any assignable cpnintity. Moreover Laplace's argument regarding the sulliciency of any particular solu- tion (lid not occur immediately in connection with tiie treatment of ilii;, par- ticular case, and it therefore appeanul that he ottered no justitication for the assumption. Airy objected to the assumption on the ground thr.t it was unnecessary and unduly specialized the solution. He added that, if the sea were boun necessary for till' determination of the second of these constants. He siiowed that, since the oscillations of the water which are taken account of in the ditt'ereutial equation have a perfe.-tly definite period depending on the period of the disturbing bodv, the original state of motion could not be taken account of in the solu- tion, for, except for sjjccial dc[)ths of water, the period of the latter oscilla- tions would be ditt'ereut from that of the formei-. Airy and Ferrel, however, did not iulmit the force of the reasoning by means of which Kelvin justified Laplace's result, and Ferrel's later papers are devoted to an attempt to show that the determination of the value of the constant is unnecessary. While it would seem that the constant was correctly determined by Laplace, it appears to the author that the analytical proof of this fact indicated by Lord Kelvin, and given in greater detail by Prof. Darwin, is not complete. It seems, too, to be desirable to obtain the general solution of the differential equation, and to follow up the suggestions of Lord Kelvin regarding the application of this solution to the more general case and some special cases. ■■', ' "T T ^ivwpqw^^Mf^^wiiqqpqQiwmnniB'ViiP^g^ ^mmmmmmmm WHICH I /lESKN TS ITSIXF IN LAPLACE's KINETIC THEOIIY OF TIDEH. 0!) TTI. Lai'i,ace'h Som'tion. 0. (n'liii'dl (infliihi. Apart from phyHiciil coiisidoratioiis, tlic iirf^nniontH inado l\v Laplacu h'om aiialysiH sine c)f the polar distjinci! of a |>arti(;le of water, '/ the dillerenc(^ between the tide hei^dit in the dynjimical theory and the tide hesf^ht in the ei|uilibrinni theory. Assuming as the solution of (i) a Taylor's series containinjj; only even |)o\vers and witli undetermined ooetWcients, Laplace found tiiat the coetticiont of a;' remaini>d undetermined. He next proceeded as follows : He assumed as an inte^'ral a sum of ii (inite nundicr of power.*, and found that by addin<^ a certiiin term to the left member of (1) this etpiation could be moditi(Ml so as to have as a solution the assumed function. By study- ing' tlu! ell'ect of inoreasint,' the number of terms in this function, he came to the conclusion that a very snndl error would be made in assuming as a solu- tion, such a function .vitli a large number of terms. (■». Apjiliciifl'in. To apply this treatment to the ccjuation (1), assume that u =A^ + (.-l, E)jr+%A^r'^. (2) If this function satisfies equation (I), the following relations must be satisHetl l)y the coelKeients : (a) (0) .1, .1,= A. 10. 1, - 10.L + ,'i^^ = 0, 2.1,+, \:2{k - D- + C (/; ~ 1)] - 2.1,1 2 {I- - 1)- + 3(X' - 1)] + ,iA,_ , = , {k =r 3, 4, o, . . . , r.) -2.l,^,(2r + 3>0 + /^'lr = O, + ,U,+, =0. (i) (g) t 1(10 I.INd. (IN Tin: HOLUTION or A < r.UTAIN DIFI'KUKN'I'IAI, KglJATKtN Tlioro iiro, rtiutio (c) iw iiii idtMitity, /• ■\- 'i liuotir «i|iiiiti()iiM to l)o HutiHlind by ?• -j- 2 iiiikiiown <|iiiiiilitii\s. Il is oiisy to hod timt all ciimiot lio Hiitisfi'id ; for, Htiii'tiii^ fi'diii (|^j iiikI \v(>rkiii<^' l)iu;k, tlitiro ritHult /oro vhIiioh for nil lUv A s .uid this (lisiii^rcds with (li). If, however, oiio of tln! (uiuiitioiiH Im rrji'ctcd, tlui rciimiinng ?' -j- 2 it'latioiis an* Hunicifiit to dcituniiiin' tlic vidiirs of Ww A'h, Sii|i|)os(i (ix) to !)(•! rcjt'cttMl. Followiiij^' riapliicf's imihod Irt the followiiif^ alihroviatious I to iiiadu : /I =-- i,i , /i, = 2r' + 3/- , /i,,, = 2(/' - 1)- + :J(/' - 1) ~ [(/• ^- IJ- + 3(r - 1)1 /////, , /I..., = 2(/- - kf + 3(/- - /,■) - [(r - /)^ + n{7- - /•)]/'//'■-*+. . Then it follows that wliciioo yl,.+, - ;:■'" /!,,== vl,._, = ■" A .. . '■ 1 — '' ' II.' A', //,//., «' I,"'.' .'I ,n,it.,it., -1,4, /«' /^l/^a,":. /':. Thoso valims of .1 1, -L, .l.j, . . . , ^l,.+, satisfy the oqimtious (a), (li), (c), (d), ((!) (f), l)\it (Miuatioii l<^) is not satisfird, and the fmictioii (2) is thl>}i i-tiiins Id iiiitliiiil. In order tlmt tlie diHcUHHion jiiHt ^iveii imiy justify tiin clioice of tlie vidiii of A.,, it HJioidd put in ((videnco hoiuo property poKKessed liy tlie series when Tjiipliici^'s ''.tlue is ^iven to .1^, iiiid not posHcssed 1. , it under other circuiustiinces. All thnt is iittenipted in the precediu}^ ]ir(tcesH is to show that the error inadt! in ussuiuiii^^ us the intej^ral a Unite iiutnlior of terms lieloni^iii}^ to the infinite series can lie nnuh^ less than any ass!<^nal>lo (juantity. ir of tfu'uis instead of an infinite! numlier. H. liiiliitiiiii liitirmii i'in')'i'i'tnin bi ii/nitfidii (Iid/ i'rr'>r in Kri'ii'n. Tho rela- tion betweon tho corroction to tho equation and the error in tho value of tho dopondoiit varialtlo can he shown j)orhaps more clearly in the following man- ner, assuming (Certain propei'ti(!s (if the series used which will luMleduceil lat(^r : The difVerentiai ('((nation (1) can lie regardcid as a linear reflation connecting tho (inantities ", , , and , „ in which the coofH(Ments ans rational entire func- ' il.i; il.i- tions of X. TIh! function k is to ho oxpross((d liy means of an infinite sorios. This s(!rics will have a ciu'tain circle of convergence. For all points iiiit/iln this circle th(> corresponding series for and ., will also i-onvei'ge. Sup- pose the circumferonco of this circle of convergence lies entirely outside of tho houudary of tho region in which the independent variable is to vary. Then when ", , , and , ., are each replaced by tho finite number of terms from tho (l.r (/,/;- ' '' series expressing their values, certain errors will bo made in tho case of each. These errors will each become less than any assignable quantity if the number of terms be sufHcicntly incroasod. Let (1) then Ih; written -, (hi. , .j I I i 1 i S. i 102 l.lNd. ON THK KOT.rrrON OV a CKIITAIN DlI'l'-KliKNl'IAl, WiHATION wlionw/, ,V„ ;-, iiiwl o iuv (iiiito for all viiliios to bo consid.'nMl. L.'t the Inui VMJii.'s ..f "' " , "'" , iiiul " wlu'ii foun.l from tl'o iiilii. Ic series l)o .1, /A aiul r'; and 1. 1 i,, J., and £, l)i' tlio (•orrcHi)oiidint' oirors mad.' in taking Cu; linito niimlii- of terms \,,r each i>f tin' Mut'c (jiiantities. 'I'licn 00 |'|„,„ tl,,' .•uiiv,-ti(.n to lieadd-d t<. the left munib.ir of {k) to make it an ,i,,iljtv is I'/s, I ,;,c; i i'-i)- I'nder ;iii/>iiicr'.: (iKsiniipl'oui. The same set <)f ,., Illations for the determination of the (^oetUci.mts will 'le olilained if it be assumed that £ A. ' ' rr^ . I'or then it w■■ proper to assunn' tiiat tli piidions (onneding the co.d'- licieiitH es.uitualiy took the sam(^ t'orn; as (f). l-'or all other se.M's in which the ,vl,ili(.ii jiist written is not true, tiie e.iuaiion (f) is not satislied when a finite number of terms is taken, but has to i)e corrected by the a.hlition of a term. |''rom this j.oint of view then liajilace's proeesH would seem to noceKSiirily lead to a series convergent over tlu! entire plane. WHICH I'ltKsKNi's ITSKJ.K IN i.ai-i.ack'h KiNirric THKoitv ()!•: 'I'lDiOH. iu;i TV. The Somition or thk DiI'Tkhkntiai, Equation. Ml. (7i(iriic(i;r nj'f/n i/iffi/ra/. Ijiipliicc^'s Holulioii having Ixhui I'oiisidon'd, tliii gcnoriil iiitcgnil of llic, luiiiatir:;, ^^"{1 - x') '''". ,' '';' n (H 2,:^ -- fix*) ^-. AK,^ , (I iiia_v now 1)0 Houglit. In this (■(|ii,\tion it nuist hv rcnicnilicrcd tliat ,f :: sin I) wlitiro (\ iH a puhir distanftt. It i.s nc^cos.nai,)' (irst to >solv(^ (he anxihary (!(|ua- ti'ui. iPn ,l,i «=(l~.;^)y .v"l'^ ,,(H 2,r ^V) -.:(), (7) Thr following g'.'Mi^ia! tlicoifniH will !»! nscful : 'rnr.ouKM I. /;/ tirdcr //iiif t/if ci/iKilidii tiiiii, Kiie/i (in /(„ the paint ,/■ (I xhiill lie an urdiniiri/ point, or ti pole, irhdse order of nmlti/ilieiti/ does not i.reeed i. 'l'lu^S(( integrals will constitute! one or -v. groups of the form "i -- .'•' J/| , n, .-.,,•'■ (.IAI.,g.,; -I a;), "3 -.'••(, I/., log-.,; I v\'., log.,' I //,), «,=--.,••■(, I/Jog'' './■ I ,VJog« V' I ...), when! .!/,, .)/,,, J/„ . . . , .I/j. dill',)i- only l»_y constant fuctors. Thkoukm II. '/'he, inte(/rdl of the ('(pdition (H) irill !„■ rontiinious und ■nioiioijenir fo' oil edliies oj' .,• _/;*/• (/'/,/,•// thi' rmj/irienls p^, p„ p.^ p^ ore eoiiliiniods diiil iiioiHK/enie, and it eon poxursr no eritiedl points irhieh are not 1 1 I I % I % 104 LINO. ON THE HOLUTION OF A CEUTAIN niFFEUENTIAL EQUATION aim critical points of om or more of the ,u>,^i/identK ft may not hare critical pohit>^ at all the critical pointH of the coeffir],'M><. „ - , # The proofH of theso tlieoroiuH iiitiy l)c) foiuiil in Jordan's Vourx l action of the complementary fanction. It is correct, then, to assume as one iuteg'-al, •v being a positive integer. Thou ^ -.- 2,. (hh -f rt<) A.^ , n l,.{>a + rs) (//. - 1 -f r,s) A,,i:"'"-+'-' , ' J identicallv. If th. substitution ,', == ^V. '-' '"-^^ -^^ '^ ^^ ^-"'^ "' '"^"^^ that «, is a soUitiou of (7), (15) roduces to ..a--)';^-'--';.;^-"^'^^'-'"'^''" ) - (2 -^ a^) ", '!- 2* (1 - a^) ^^ = . (1(5) It' now it be assunietl that and th is, as before, ^'v, = 2', /A ■'•■"+". ^^''^ . substitution bo nnule in (IC), the equation for the determination of m } _ 2»/« —8 = 0, whence m = 4, — 2 . VI' The theory shows that //* - — '^ is « = 2. The substitution in (IG) gives 2 is the root to bo taken, and that, as before, iV (2r - 2) (2r - 3) //..r'^^ - %■ (2^ - 2) (2r - 3) T/,.^^'- - 1, (2r ^- 2) B,^:^-' - X « /A-r^ -^ + f 2 /A- ■'-'• + V 4 (;• + 2) (\..P-^' - iV 4 (>■ + 2) 0^'+" = , in which the r's are the coefficients in the series «,. (18) ' LI WHICH PREHENT8 ITSELF IN LAPLACE's KINETIC THEOKY OF TIDES. 107 From (ID) are derived the relations =_- 8/A + 4Z/, , = 8/;, - 2//, 6 = ?Ji., (19) 7 - lor; = iG/;, - 10//, + ^n„_ , {U - 5) (\_, - {ik - 2) (\_, = {2k - 4) {^k + 2) //,+, - (2^' - 4) (2/J- - 1)11, (k = 4, 5, G, . . .) These equations determine all the coefficients in terms of li^, the coefficient of x\ wiiich remains arbitrary. Since u^ is a solution of (7), it is wear that, if if is a solution of (IG), so also is f + yi',?<|, A'.^ being any constant. Moreover ;/, starts with the fourth power of X. Then if any value be assigned to //, and the resulting value of ^''3 be denote.l by '., may, if it is desired, be taken to be zero. The com- plete complementary function of (17) then is '' = ^I'l (v''2 + 'I'x log x) + A'.^i, = ^\ {'/•2 + ^l'"! log x) + A'^%, = ^I'^-l'l (V'':. + "l log X) + A\,H, = A ((f A- n^ log x) + Ba, . (20) 12. The jmrticuhir integral. It remains to determine a particular inte- gral of the complete equation (1). The character of ")] A„+.,/A„+, (21) nives th" value of A., in the form of an infinite continued fraction. It is per- missible, if convenient, since any value of A, may be taken. It was made by Laplace in the Mc^-caninue C.'^este apparently without justihcation ; but, as has been seen, Laplace believed in the sufficiency of a particular solution and considered the resulting series as a satisfactory solution without the addition of the complementary function. The assumption (21) so affects the coeth- cients that th. .cries converges for all finite values of .r. It is not necessary that the parti.-nlar integral shoul.l converge for points outsi.le the unit circle. It is convenient from a mathematical point of view, however, to choose this series as the particular integral, for, if it were necessary to study he function for points outside th . unit <-ircle, it would be sufficient to obtain the cx,mple- uu.ntarv function in the form of a Laurent's series while the series just found would ;.n-ve again as a particular integral. The assumption (21) i. seen to be ecpiivalent to that involved in Laplace's original proce.ss. From the reasoning of this section it is clear, too, that when £ A., then Xi A„ 1. it WHICH I'1!i;si;nts ri'sEM' i\ r,Ai'LACK.s kinetic theouy of tiueh. 10".) From tliis ))oint of vi(!\v, too, it is (iloar tliut tlio sorids under coiisidfriitioii < oiiverges ut ItuiHt for (tvory jjoint wilhiii tlio unit circlo, mid tlmt if it convcrgeH for II groiitor circlo of coiivergenco it coiiverj^cH for overy fiiiito viduo of u-. For tlio Kfiko of defiiiitiMicss Liiplaeo'.s viihus of A.^ will ho donottul by /- and tilt! serieH which furniHiios the pjirticular iiit<'<5ral of (1) will ho denoted hy V. I'ii. I'mpcrtien of the t'ompleU'. iiil''(ir(il. The (•oini)lete integral of the equation (1) ean then ho expressed for i)oiiits within the domain of the origin i»y u = A (if + u, log ./•) ^f 11 „, 4 V . (22) (1) Convergence : The most general integral in the form of a positive power scsries ean lie writt(in a = B */., 4- V . The relaticiiis among the ('oetHcieiits of such a series sliow that, unless II is zero, the circle of convergence is of unit radius ; and when // is ztno, the circle! of eonviu'gence has an indefinitely great radius.* It follows, then, that ^/| converges only for points within or on the circh; of unit radius. Again, y''., = ^ + A. ,11^ does not coiivesrge for points outside the unit circle. Then if converges for points within the unit ci'cle. It is conceivaljle that //j may have been chosen so that ^ shall converge all over the finite part of the plane. (2) If A -- li ^= 0, the function has no critical point exce])t at infinity. If A .^ 0, /> 0, the function has critical points at ± 1, ±: x . If II — 0, A -^ 0, the function has a critical point at 0, ± co , and (except for one ]iar- ticular choice! of //,) at ^t 1. In addition to the singularity of ^' at ./■ =; integrals of this class and integrals of the general class have a singularity at ./• = due to the singularity of log .'■ at that point, and are, in addition, many valued at any point owing to the properties of log x. This iiideterminateness will he removed if it is assum(!d that for positive real valr.es of .« the result shall be real. Again, when // =~ 0, But ilx 1 r^/tf , , i/ii A I ~J- -t log j: , L 'Ix h; -r 1 1 £[; It) ,1,1 — ~j- I 1 — x^ , J 9= ^ dx -i |(log,c) I 1 dx * The (Ictiiiloil proof of this fnot is (jHoted in Section VI. Y it j . 5 «1 110 LINO. ON THE SOLUTION OF A CEHTAIN DIFFEUENTIAL EQUATION It will afterwards be sliown that for the complete integral (22) a tinite quantity, aud that J »=»/» f^^O = a finite quantity. .•. for all integrals of the form n ^ A (f -H "i log ./•) + V , [ "'" 1 = a auito quantity or zero. [,I0 '■ ^ I 9^ir/2 V. The Determination of the Constants foh Laplace's Case. 14 Tin- pinjdml coiuVdhn,. It having been agreed that the constants shall be determined to suit the boundary conditions, the case discussed by Laplace, where the whole earth is covered with water, may new be treated. The expression for v, as given in (22), has an infinite value when x = 0, unless A = 0. Since there cannot be a tide of infinite depth at the pole it is neces- sary to make A = 0. The remaining expression is n = BUi + I'. Airy and Ferrel contended that this was the exact expression for u, and that JJ could be giv n any value. Ferrel determined it by the condition li+L^O. (23) Lord Kelvin poiiUed out that owing to the symmetry of the disturbance in the two hemispheres the meridional displacement of water should vanish at the equator. The expression for the meridional displacement is the product of two terms of which one involves the latitude and the other does not. The factor involving the latitude is, (24) s = i)ii ain^O ■^« + 2./cotw| When II = r/2, Am. iEH. at the equator it is noeossary tliat (tti dd = 0, 111 (25) But •. it is necessary that, mi -. , fhi I 1 — x' -, . £1'''-J - 0, tho integral of (28) cau bo exprosseil in the form ^\'J\ + Then Also (la (IVl ^ ,,, "yi .,. " dy., e.piator dn/dO - 0, tho function must be such tliat <■, - 0. 15ut if '■, -^ 0, tho point ^' = 1 is an ordinary point of tho func- tion, while if <-, the point a; = 1 is a branch point of the function. If the point .'• = 1 is an ordinarv point of the function, so also is x = -- 1. In this case th.> fun.-tio. has no crili.^d point in tho finite part of the plane, and, if expressible in ihe neigliborhood of tho origin .'• = by means of a Taylor's Hories, that series will converge for all finite values of the varia- ble j:. If the point .'• -- 1 is a branch point ..f the function, and if the function is expressible in the neighl)orhoo.l of the origin ./• = by moans of a Taylor's series, that series will have a circle of convergence of unit radius. It follows. rt WIIICII I'llKHKNTH ITHKI.K IN I.AI'LACK'h KINETIC TIIF.dllY OV IIDK.S. 1 Ut tiicii, wli, I />' {) iuid u V, tlmt (.. = and (*/« '/")»=it/'j -- ". i>ial when // ss tlnin unity // is ne>^ative, and y* is a pure imaj^imiry. The <:.)/■ will he real when Co is purely imaginary, and it follows that then //., is also veal. VI. DaUWIN's PllESENTATION OF Loltl) KlXVIX's PitOOl' THAT li MIST JiK ZEIt(J WllKN 1(). //(ti ii'!/t\s iiiyiniii'tif. The function u -~ />»/, i V may lie regarded as u single series of even positive integral powers commencing with the fourth and having tlii^ coetlicient of .'•' arbitrary. Tt lias already been seen what rela- tions eonniH-t the ct)cllicients and detine them in terms of tlie coelHeient of ./•' {A.2 say). It is known, too, that when £ . -''•ii+i = then A, = L. Suppose now that £ * A A "+' 0, but = «""', a finite quantity. Then A„+,_ 2n + 3 ^l..+i /i A" 2n{2)i + 6) J„+, 2n + 3 I'i 2ii + G '2« (2)1 -f- ()) T-, (« - /') (34) IPUPVHP Mi 114 t.INd. OS rilK SOMITION OK A ( T.UTAIN DIKKEItENTrAI. EQUATION ' whero /, tenuis to /.to wIum. >, »..-comcH iiuletinitoly Kivat. D.viwii.'s ai^iii.ionf XH alon^ thu followiiin linuH : NVlifii £ ■ ^»+' >. tlicn, for liii^'u vahu'rt of /^ ,.^,a!-^'+'^ ~ 2«. + «} •' [' 2(«. + 3)J ^ 2// J ar U''ii>'ly- ,. , ,, lint if (1 - .'•')* Ijo oxpamlod by tlio Miioiiiinl tlu^orom the ratio of tlio (« + l)th term to the >dh term Ih [ 1 - ,j*^^ j ar. Conse.iuontly, in thin cuho, it irt i)OBsil)le to write ' „ = .1, + //, (l -x")», where yl, aiul A', are finite for all values of ,'■. A similar argument beinfj; made in the case of the Heries for iiii/ are finite (and not zero) for all values of x. 15«t ., , ,. t/„/,/ii - ii,i/U, = ^^ ' « • 17. jyiscnssion of Darx-in'.y proof. Tiiese results ap-ee with each other, and witli what has been proven in another way ; but f.as proof of the faet that //, and /> are not zero nor infinite does not appear to be entirely satisfac- tory, and it is esHoutial that this property of />', and /> be nnide evident. The ratio /l„+2/^«+i l>ecomes 1 - i? ;/-' when the s.pmro and higher powers o^ n ' are neglected. If after a certain value of n .piantities of the order u ' be neglected, the ratio .l,.+,/^l„ beconu-s unity. Tlien, foMowing a line of argu- ment similar to V it given, it would appear that „ = A, + B.,{i -^T' (*) and, by a similar course of reasoning, that dii/,h; = (', + A(l - x-y. These results do not agree and are incorrect, but they show in what respect WHICH I'liKsi'.NiH iTsK.i.r in r.AiT.Acr.'H KiNF/rtc tiikohy dk tidkh. 11.- tlif iJifvioiiM rciiMoniii^ Ih wciik. For, Hupposo tlint tlin liiiioiiiini oxpiiiiHioii of (1 .'•-)' Im written Tlit'ii, if the intinito Horien can !>r writtt'ii in tlu> form » == J, -f- /;, (1 -^■')i, wlioro .l| mill /i\ ans tiiiiti! for nil viiliii's of /', it follows tlmt • .1, "■^i-l:- Now for over}' fiiiito valiio of ti {^'i \nni\^ ponitivo), Diu'win liiiH sliowii tlmt A. ' r. (b) l)nt it is iiocossiiry iilso to show tlmt i V .1, .1 A,- A, .1, J, +1 (0) is tiiiitc. Both miriifnitor iiiid (leiioininivtor aro zero so that the value of the ([uoticiit retpiircs iiivcstif^iitioii. The tlisoussion of (a) and (i;) shows tlmt in (a) A, and //. do n()t have finite uon-vauishiug values for all values of x. VII. 18. Ciitii's to hf ti't'iiltd. It remains, then, to exandne the other cases included iu the solution obtained. Airy i)ointod out that in the solution of tlu! form u= liu,(.v) I V (.'■), // could '■■> determined so that the solution would l)e suitable for the oas(\ of a sea fonnin^ a splu^rical cap and extending,' from th(^ pole to au arbitrary parallel of latitude. Lord Kelvin jiointeil out that, if the general solution were t\ .j;^ NPPI 1 \(\ i,i\(i. OX ruv, son TioN or .\ ckiiiain pirn;iiKNri.\i, i-'.irAV)oN lit hiitiil, tli(> two rnnslinits coulil 1m> ili-ttMiiiiiic.l so iis to olttiiiii ii solution snit- nl>l(' for II zoiiiil s.'ii Iviii^; lictwct'ii two paiiillcls of liititiidc. 'I'lio most iiitri- fsliii^ ciiscs to l)t' (Iciilt with a|)|M>iir llien to lu' III.' foliowiiig : — 1. r./,sr (>f''t s,ui CDiyrliK/ the wholo <,irtli. This is tin ntseitlrMily treated '2. Cixti of It si'it. <'.vtyn>/iii>/j'i\>iii //'.' />"/'• to " '.I'n't'.ii /uwillcl «/ /,itilii(t('. 'i\. (^iiKf. of it ZKiKif ,V(W lioiniilr,! hij tiro />iir,i//<''s of Itititti,!,' on oj>ji(Wite tili/eK of the ei/ii>r/or mol rt/ini/ii/ ilisliiiil from it. I. ('is, ,if (I .dim/ st'i hi, moll s(>a fxtciids oiil.v vO ii ^nv(-ii iiariillcl of liitiliui<' from 111.' i)ol(>, it is •ii'.'cssiin iliiit tin' nui itlioniil coiiiiioiicnt of llio iii.)tioii sli.ml.l v.iiiisli for 111.' corr.'sii.iii.liiij^ v.'iln.- . f r, llic sin.' of lli.' |>o!,'ir .lisliini'i' of llli' I-.iiUIIiImI'V. 'I'll. 'II. as in Case 1. .1 0. and 111.- condilioii just nam. '.I ^iv.'s for tlic boundary value of .'• fin it // „ 0, Ij.'I. lli.'ii. ", 1>.' Ill'' ■•ol;'liliid.' of 111.' sou 111. 'I'll lioiniilai'v. an.l sin ", -- k. (Ill I'll II , an.l i.'s II, ,ln 'In (l.v sin r^ (I, f..r l> -- 0, /!o;{u,) ^ v'(«,) , " ;/.'",(",) i vc/,); - o //-- «,V'(",) I '-JVC/,) ",",'(«,) f 'i"l("l) 'I'liis {^ivi's i\\v cxprossioii for u at anv |>oinl in tlif f.nni = V(.'') ,/.,V'('/,) I '2V('/,) A-'') In jiailioular, for tlic souIIi.m'u li.Miiulan, '(«.)- {•.\r,) :i(i) (37) wiiuii riiKSKNi'M ri'si',1.1- in i,\i'i„\(r. s kinktic ihko'iv hk rii>r.s. 117 (lil') inul {'M) ^;iv(< tlic itnioniits to !i(> iKiilcii to llio tide ili'tlnct'd from llic t'i|ni- lil riiim llicory. Tlio toliil ti !(• ill any jioint lli 'H is f =^ «,(,,■) , /'.'.r , iiiul lit tlic souIIh'I'ii Itoimdiiiv /' =.- II («,) j /•'u^■ ,:i8) ( :!'•>) Ah liiis lu'cn st.'itcd. l-'crrcl's ciilciiliitioiis wcir in.iilc for llic series in wliicli />' wtis ji;iv('n l>y I lie rrhilioii /.' / 0. 'riiiMi, from (."t.")) it npiK'ius tiuil tiic tiilcs imIi'iiIhIciI by I'crrcl would ix- (Iiomo cxisliii}; on !! circiimpoiiM' scm lionndcd by ,1 |>nridl('l of latiludf ,J~ ll^ wIhto a = Kin l> iinil sulislics the riuialion /; iiV (11) I '2v I") Cask :{. ('.Whf) ■JO. Si, I < ri, iiiUiiii , ijiKill II :>ii Ih'tli sill, n 'l' , i/ii,il,ir. Siipposi- the si'.'i to cslcnd i'i|n;dly on iiotli siil<'s of llic- ('i|UMtoi-. ||ii> lioiind.irit's IxMnj; jiarallcl^ of latitndc. 'I'lio condition lliat tlicrc sliall lie no motion of watiT aloni; llic nu'ridiiin at any point of tlu> nortin'rn lionnilary ij;iv(>s one relation coniuu'linL; .1 and /•' ; Init it is cli'Mr 1 1ml the ('oir(>spondiiii;' eondilion for t he southern iioinulMrv L;ives exactly the same relation ; so th;it one of the constants appears to l>e arhilrarv. The considerations wiiich applied to ( ';ise i apply to this case. The s\ ninielrv of tlic motion reipiires that theie l>e no meridional motion of the water at the eijuator. In this case, also, it is necessary tlnit This [fives a second condition In means of which th(> remaininu iirhilrarv con- slant may he determined. l''iom >i .1 W I ", l<'^;-,fi I li», i V {'I'D it follows th.d .In ,10 .1(1 —.»•■■)» (f' I */,'lo^'.c I ^ //,) I /.\1 ,(■-)»«,■ -f (1 -,/■••)» v. M 118 LINO. ON THE HOLUTIDN OF \ CEISTAIN OIFFEItESTIAL EC^UATION Now £«- ,7"-)4 V'J = ; r(l - .7r)l -^ «, 0. since «, converges for .r = 1 ;* _£[ (1 -*•-)'",' loj^ ■'■] = f'. siiici! log 1=0 and J^[(l — «;")■ "I'l i*^ '""^'^• It is clear, then, tii.it vl £l(l - •'.■:)^ ] = '> • It has beea seen that 4- [(1 - •^")' "I'l i^ ^ *i"it ; -J- 1(1 — *'■)= v'l '** '^ ^^"'''" - . (40) Returning now to the condition first stated, let «, lie tlie colatitude of the boundiiry, and lot sin «, == w,. It is necessary that since cos "; ' 0. The resulting relation between A and /> takes the form + yi|»;(«,)+ '^",(",)i-i- v(«,) , 'fv(«,) = «. (12) I J ♦ Hec eciuiitiMii ( b) Section VI. The I'xpaiiHiou of (1 — x')l fouviigcs for x = 1. L^^ WHICH PliESENTS ITSELF IN I^VPLACE's KINETIC THEORY OF TIDES. Hi) An ctjimtion for // is obtainod by uliiiiiuntiu}^ A and />* from (22), (40), and (42). Tlie total tidu is Case 4. 21. Sfii houiidi'il hij tiro paralleli) of lat'dmle mi flw .sdiiw side of equator. Suppose the sea to l)e bounded on the north and south bj- parallels of latitude and to li(! entirely within one h(^niis|)here. It is necessary tluit, at the northern and southern boundaries, dii 2 dx X n -= Let tii(! boundaries have colatitudos II ^, 11.^ (H^ r' H.^), and let sin II ^ := «,, sin//o Then the e(juations tor tiie deterniiuation of A and li are similar to (42) and are r 2 12 1 A I « «i + "r(«i) ! , "i("])log«n"i'(«i) -!- , »m("i). ^A'"i) ^- , ''('/,) n 1 '2 2 2 tf'(«,) + V'(«2) -I "l'(«-..) log «, -4- ".("i) + "l("-.01og«,, 1','("..) -f , ■"l(«2), ^"(«2) ^ „ V('/.,) 9 19 2 9 12 '' ?< . (44) ! I 120 UNO. ON THE SOLUTION OF A CEUTAIN DIFFEHENTIAL EQUATION (•14) gives the value of u. The comp'ete tidal expvessiou is u -f Kur . Till) value of n at the bouiulary whose colatitude is t\ is given by 0. . y «?'(«,) + " f(«,) I- »'(«l^l«g«l + "l(«l) + f "■(«■) •"R«H «i'('«i) I ^t "'^'''') 9 12 2 tf!'(«^) + " y^K) + «'K) log '-^i r "iK)-l- '» — I i J (N + 1 ■^ ;^ 1—1 ■§ (MlS ■ 5^ 5 _ + + ; a ■-C ^ *<" + V 5 "^ 2: ■r <0 C ^ ^- tc f -V. ^ -: " U) 0^ ■' — C s s ■a /■-*, , *M g v£ 1 « B (N jT 1-H "-■ ^ g' + + « 5? ( 1 1 s J3 *<" ^T" a a " ^^ ■ ^^-cC Sm V f-i 5 •C i X ^ to 01 ^ + ^1 V 1 H ^ ^^ ■"^ fs 5<" ^ ^ ^ M •««j (N !> (M ;?■ ^ f* 1 ' 4- tc -j- 1 ^ _0 ^r ^ *s* + u. n »- y "u^ (>j a" «^^ 5~ o^ I a" — ^1 sT cc e tc tc ■^ a tc + + 5a- + + V + Oi t^" 5" eo i a" — - i- "J- ^j- (Mi',r tc + 122 LINO. ON THE SOLUTION OF A CEUTAIN DIFFERENTIAL ECJUATION 23. Title III j)oint dintant d from hoamliiry of canal. For any point within till) canal distant i) from the northern boundary x — a^ -f ii y'\ ~ a^. This siihstitntion beinf^ made and the product ".'(«.) + I ".(«■) , V'(«,) + - V(«,) •'i "■{ "■\ a, a. f(«i) , «,(«,) ^'(«.) ^ — Wi log «, ?», f'(«.) + - «l(«l) . '(«.) , ^"(«.) + ~ i'(«,) '*1 "l '*! Ui li^ V'(«l) + I f («l) + ■"■- M,(«,) «1 «! . "l'(«l) + „ "l(«l) "1 '*1 «i «i + u I, 1 ~a^-i\u,) -f- * ,/,(,/,) , n = V"'(«0 + ^ V"(«0 - f, i"(«0 + \ V(«,) . "i «i «i / WHICH rnKSKNTH n.K.K IN ..placb's kxne^c THKonv or riOKS. luination (48) can then be put i.. the form 123 f'K) H^fK) rj;«i(«.) . «»'(«») + «,"'(«'^ .>o + ^x«o<"/(^o + J«.i«o.<(.) + ^'''(«o «, 2 ..„ + '/ > f («i) _ V^'(«.) + «, "'^''') ,<(«.). n«.) fi+''^' ''V'^^^' + 0) 2 V \ ^ ,1 (a) »,"(«i). ^^"("i) 124 LINO. ON THE KOL'J'J'ION 01' A CEUTAIN DIFFE/iENTIAL E(K'ATION As bffore, tlio totiil tide is VIII. (51) SlMMAIlV UK HehULTS. 25. SinniiKiry i>f f-/V. For coiivtsiiieiit'o of icfoiuiico, and in ordor to riuuUr tlio resnltH available to any who do not dosiro to follow tiirongli the proi'csscH of ohtainiiif,' them, it has been thought desirable that tliey should b<' restated in a separate section which, along with the historical sketch in Section 11, would give a complete account of the state of the i)roblem. Sec- tion III is devoted to a discussion and criticism of the analytical process by means of v,hich Laplace obtained his valua for the arbitrary eonstaut iu his solution. Objection is taken to tlu; process i-miiloyed for two reasons. Tu the lirst place, the reasoning used has not been shown to be, and docis not appear to 1)6 strictly accurate. In this connection it may be said that iu the oxami- Uiition of the iijjparent inaccuracies it has been thought sutlicient to indicate the weaknesses of the method rather than go into a minute discussicm of them. The modern advances in the theory of DilVerential E.puitions make it appear prol.abhi that matters of this character will be treated ditlerently in future, in Section IV the complete solution of the e(puition is found, the expressions involved being infinitt! series, whose r.'gions of convergence are large enough to m.ike iH)ssil)le the treatment of all cases that can arise. The regicms of convergence of the series tog(!tlier with certain important properties of the integrals can be |)redieted from the form of the tupiation. The integral fouiul is more general than that previously deiluced involving the two ari)itrary con- stants. Th(! series used by Laplace (inters this integral as a part of it. In the derivation of the integral the method of Laplace as given in the Mecanique Celeste is made clear. In the closing paragrai)hs of the section certain pro])-