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REPRINXED FF.OM
- ;fi\,^j.
\^^■r
ANNALS OF MATHEMATICS.
•^-.
Vol.. X.
April, 1896.
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 IjIN<i, New York, N. Y.
^ TABLE or CONTENTS.
A ■
Paiik.
.\it. 1. Objects of the paper,
I. IntboductohV.
II. HiHTOKicAi. Sketch.
10
11,
12,
111
14.
15.
10.
17.
18,
1'.),
20,
21,
22
23
24
25,
2fi.
Origin of the prohleiu,
Previous ooiitrilnitioim to the subject, .....
Synopsis of previous contributions
III. Laplace's Treatment.
General Outline,
AppliciitioM, ......••••
Objections to the nietliod,
lU'iiition between coireetion to the e(iuiitioii luul error in series.
.\ssnniption eiinivalent to Liipliice's nssuniptiou,
IV. The Solution oe the Equation.
(.Ihanioter of the iiitei,'riil. ...••••••
Deduction of the coniph'nieutiiry fuuetion, • •
'I'lie piirticular iutegriil
Properties of the complete inte^jra!,
V. The Okteiimination of the Constants fou Laplace's Case.
The phy-iiciil (■onilitions,
Proof timt y; =
VI. DaUWIN's I'llESENTATION OF Loill) KeLVIN'h PROOF THAT Li MU.ST BE ZeRO
Darwin's argument,
Discussion of Darwin's proof
VII. Application of General Integral to Other Cases.
Cases to l)e treated
Polar i
Sea exteudinfi eqiially on l)oth sides of the ecpiator, ....
Sen liounded liy two pandlels of latitude on the same side of the equator,
Canal of width 2(nyii.i.,'ali>n>,' 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- <lij rAoadumie royiile do
Puris, uuut'e ITTo C.
1
T
W IIKII I'ltESENTS rrsFT.F IN I,AI'LA('K's KINETIC TIIEOI OF TIHEH.
'.•7
3. J'rerioiiti contrihiitiunx to the Nn/>}'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<led by a paridlel of latitude instead of covering the whole earth,
then the arl)itrary constant could be determined from the corresponding
boundary condition. Ferrel agreed with Airy, and regarding the constant as
being entirely at his dispo.sal, b,.sed his calculations on the series resulting
from assigning to it the value zero. Kelvlti in his replv to these arguments
quoted Lai)liice's reasoning regarding the sufficiency of a particular solut
but pointed out that this reasoning was not correcit_J2i:ucciHWH5*riffrcoiitended
that, as denii.nded by Airy, there was a certain physical condition to be satis-
tied, and tiiat this condition was sufficient to justify Laplace's result. He also
pointed out that tiio general solution of the equation should contain two arbi-
trary constants, and that a further boundary condition would b(> 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 <lo not appear to ottor a very ^ood rcaisoii for
liis (tvaliiatioii of tlic coi'tHciciit. In order to hIiow tliis, it will Ix- iit'ccssary
to ^,Mvo Lajjlacc's disi-ussioii as it appeared in iiis orij^ina! nieiiioir. The form
of the ecpiatioii as treated by Darwin differs slif^iitly from that in whi'ii Laphien
used it, hut no essential ditVerence is introduced hy the chant^e, and tlw^ same
dilHcrulties arise in Itoth cases. As Darwin's form of the e(|uation is doubtless
that in wliioli the wiuatiou will henceforth bo studied, it has beeu adopted
here. The ((piution may then l)e written
x'(\
(8 — ±,r - ,^,,.') „ -f A;i,y'
(».
(1)
It is to be noted that .r. is th(> 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 th<?n'for(! not a
solution of (1). If, however, to the left luoiuher of (1) lie added the (juautity
— ,'iA,..^.iX-'"^''', eqiiatiou (g) becomes the identity
/!
WlllCir I'llEHKNTH ITHKM' IN I,AI'I,ArE'H KINKTIC TIIKOUY OK IIUKH.
101
HO tliiit clio t'xpreHHioii (2) in ii solution of oquiitiou (1) tliUH iiindituHl. Tlio
DOW e(|aatinii ciiii Im \vritt«!ti
^ (1 - •^) ''!". - X '{" - n (8 - 2a^ - ,^x*) | Kir" - ^"'' ,/•-'+" 0. (3)
i/.ir (f.n ' fi^fi.^ , . . /I,. ^ '
TjiipliKM* thou iirmicd tliat if tin- (iorri'ctivo tniiii wcm'o vory Hiniill, only a
siniill orroi would In- iniidd if (1) wnro nipliUMnl hy (I{). His tlicui procefuk-d to
sliow tiiiit iiy tiikiii;^ /' iiii'^c (Miouf^li tlio corrective term could lie iniidc to
decreiise ilidelinitely.
7. < >l>}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. i<ut this is true of any s<!ries which is an intej^ral of the (Mpnitio.'
and whoso n^j^ion of oorivor^onco iH larj^o (Uioiif^h lo suit i\w conditions of the
proliloin, and it will appear that no matter what value ho f^iven to ,1^ tho refj;ioii
of (•onverf:;ence of the rc^sultin^ s(!rics is still larj^e enough to suit tho |)urposo.
Moreover, it is not definitely proven Unit a vanishinj^ correction to the e(pni-
tion nocoHsnrily indicatc^s a corresponding vanishing of tint error due to tho
assumption of a finite numl)t>r 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 ;ii<! eircnmslanci^s assumed al)ove, in
»'(
n^feivnee, to tlie re^^if.n of (•onv.M-fi;ence, tliis correetion will heconn! indeliniti'ly
Hn.all when Ih.' muuiIkt of tl e terms is snlVieientlv increased. Hut oilier eases
,„,iv ocenr. Consid.^r the ease of Ih.! inlinile serii^s in whieli .1, is given any
„ti,cr vahn^ than that assigned by Laplare. it will afterwards appear that this
series converges for all values of .-• within th.' unit circle, and also for ,/• =-- i 1.
The series for '{" and "',"'! , liowexer, art nv.'rgent only for j.oin.s within the
unit .•ireie. it is clear then that, for all valr.es of ./• less than unity, £„ £„ and
£, can be made less tiian any ussignabh .iuantily, and that therefore th.^ same
is true for «£,, I (V,:,,, | ^s...
lint ,'■ must i)e considered for all values up to and inchuling unity, and it
do.s not appear thid, as r app -oaches unity, the correction must necossarily
i„delinitel\ diminish, siiH'c i, is the only on.' of the three .piantities which
mdclinitely diminishe: . .Moreovr the s.-ri.'s is a satisfactory solution if only
c, ..,in b(! made imU'ti.dtely small for .'• .1. On th<'. other hand, it would also
a,l)pi'ar that the < orrection migiit i)i' evanescent wht^n £, is not so.
'.». .[s.siiiiij'ti'iii i,iii'ir<ihiil In l.>i/>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<iuld !>■■ 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
t<hiill /id re II iiiilijH'Hil, lit iii/ci/ra/s of th, fiiriii,
n-.*-(/^_, log" '. I /^,^.Jog" ^,; I- /^, Jog" Iv
ihi
(1.1'
r„<|^A^ (H)
/', i"K.'' I /;),
ii'ln re /'„ /', /'^, /',, . . . , /'^^ I are e.rprexnihle in, (lie ihiii/ihor/nxnl of x ~-
/« seriex of potiit.ive tiiiil iieijai'iee pinners of .r, fhe iniviher of iiei/dtive poire.rs
111 eiie'i heiiKj '•'ii'if,'. It is iireessd r;/ n ml siijlieient t/mt for i mli nf the eueffieientH
(i_t the i>iii<itii>ii, 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 <r ivnalyse.
If, then, equation (7) be put in the form (8) ami these theorems applied,
it is clear tiiat, , ■ ,, , «
(1) Equation (7) has on. intc^^ral wliich .^an be expressed in the form of
a pow(!r series ^■, and a soeond int..^ral whicli can be expressed in the form
^V, + il', loK X, wliere c'', and v'', uro expressible as power series and V^; -= a con-
stant c (when c = 0, the second intej^ral is also expressible as a powei series) ;
(2) The general integral of (7) c^an have no critical i)oints except at 0,
± 1, ± 00 .
i'.h If ./; be replaced hy - *, the e(piati()n remains unaltered; whence it
follows tiiat tlu! function has tlio same character at x =- — a as at x = 4 «,
and, in paiticular, if the general integral of (7) has a critical point of any sort
,^t ,;.' = + 1 it has a critical point of exactly the same character at .'■ =. — 1.
It follows, too, that if U is any particular iut.^gral of (1), the complete
integral is , s rr
u = rt,v'' + '1,{,/k, + c<i' log ,/!) + ff,
when c --- 0.
11. /)e>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:"'"-+'-' ,
<l-a
<lx'
The result of substituting in (7) is
v_, (;,, -f- rs) in, - - 1 I r.v) 71,,/;"'+" - I',. ("'■ + rn) {m
-1- I;2vi,y"+-+'M i',.M,.c"+'+" = o,
— 1 4 /■.v).l,,X''"+''*"''
(9)
♦ Vol. Ill, Arts. 14«, "J2, ll«.
^1
WIIK'II I'liEHENTH ITHELF IN LAI'LACE's KINETIC TIIEOUY OF TIDES.
105
identioilly. Tlio cocfliciont of x'", tlio lowest powttr of x, must vaiiiHli ; wlienco,
wince A„ 0,
111'
'ini, — 8 = 0, aud iii = i ,
Tlu! valuo of .V iimst be 2, for if .s' 2 the coofticiont of 3-'"'+-(loe8 not vanish.
fhv result of snhstitutini,' .y =^ 2, in = 4 in (0) is
i', (4 + 2/0 (3 + 27') A,ar'+' - i, (4 + 2r) (3 + 2/') .l,,.y-'-+"
(I
— i', (4 H- 2r) A ,ar'-+i - i', 8/1 ,,-f-'+'
(I II
+ i',. 2yi,„Tr'+" + i; /9yi,xc='+" = c . (10)
From CIO) the e(iniitions for the doterniiuation of tlic coeffieionts are,
(11)
1(>J, - 10vl„=-- 0,
2^ (2^- + G) A , - 'J/' (2/(' I 3) .1 , , , i^ ,Jvl,_, =-- ,
(/!• = 2, 3, 4, 5, . . .) .
Tlius each cootlicieat is a uiuitiph) of .1,, and one integral (;an l)e written
yip -i|, ^i|,
= a.'' + (y H *' V + r:,«"' + . . . .
(12)
(13)
It is aasy to vtuify that A', */, is the series written (h)wii by Airy in his
article in tli(! I'liilosoplucal I\riij,'a/.in(( as the correetion to Laplace's series, J\\
being an arbitrary constant. On applying,' to the ecpiation the tlieory set forth
by HetVter,* it ap|)ears that there is no series corresponding to tlie root
III = - 2. Till! second integral is then obtained in tlu^ form
I'loni this
"•2 = ^''2 I v''i loK •'• == ^'-1 f- '1' "i log X .
iIk.. '/v''i , i, 1 (III, , A'
dx\ (Ix " (tx X
iP
i-i _ ""v
da? ih.
J +
2.1' <A,
.1'
+ A'Xo^x
</-ii^
ilx-
' Liuc'irf Dill'ereutiikl gliMchuup'ti, § Hi.
?1 V
106 I.IN-0. ON THE SOLUTION OF A CEUTAIN DIFFEllENTUL EQUATION
These values being substitnte.l in (7), there results
a?
_JV, (2-.r=) + ^l'-2,/!(l-.'-)'^^
.. ,J
+ A-Xo'^x ,j;-(l - .'. ) ^/,,. ,/,,, '> ' 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 <f we can write
^ is a series of ascending entire powers starting from B^x-'- in which every
coefficient is known. />'., 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 </, and of the absolute
term of (1) makes clear the existence of a particular integral expressible in the
form of a series of positive entire powers. Assume then
U .- A, -h (^, - K) x^ + i' A,.x-'- .
After substitution in (1) there result the relations
^„ = .
2.4,+, r2 {k - 1)^ + G {k - 1)} - 2.1, {2 {k - If + 3 {k - 1)} + ,?.4,_ , = .
{k - 2, 3, 4, . . .)
■H t
;if '
111 1^
108
UNO.
ON THE SOLUTION OF A CF.ItTAIN DIlKKIiKNTIAI. K(H'ATION
All the coofficiouts ure «iven in terms of h' mul .1,, the latt.a' be.n^ arb.trury.
Since ..nlv a partic-nlar integral m re<,uirecl any vah.e n.ay be given to A In
,„.oeee.lin\r to a choiee of a value it is interesting to follow the n.etl.o. by
'vhieh Laplace's continuecl fraction is obtained. From the relations ^vntten
just above, it can easily be deduced that
■A ,+, __
A, "2(2/'-^ -I H/')
2(2/'- -h ()/O^UW '•*■+'
(/• =: 2, 3, 4, . . .)
A.. fi
'■ 2;~ 2(2.r-: 3.1) -
(2.1^-! G.1),V
2(2.2- ■: 3.2)-
|2
2 12(// -1)' + '^("
-1)1- 2|(2//^ 1-
and A , = /'•
The MSSHnii)tion,
J^ -^1,1+1
(2.2--. n.2),i
2 (2 . 3- H- 3 . 3) — ■ ■ ■
2(,, _iy 4- fi ("-!)} /^
3//) -(2//M- <>")] 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. i</tij g,„/i
* = co-latitude or polar distance.
,^
WHICH I-nEHENTS ITSELF IN LAPLACE 8 KINETIC THEORY OF TII>EH.
at the equator it is noeossary tliat
(tti
dd
= 0,
111
(25)
But
•. it is necessary that,
mi
-. , fhi
I 1 — x' -, .
£1'''-</:h«
(20)
15. Proof that li = 0. In order to prove that tlic ecudition (2G) requires
tliat B shall be zero, the function will be considered in the neighborhood of
X = 1.
Assume
x-.-=\^y; yil)
then
(lu (lu (ihi d-ii _
Tfx ihj ' <l,ii' dy- '
and equation (7) takes the form
, (I'-u
dii
(22/ + !Sf + 4y' + 2/') ^t^ + (1 + V) J^^
+ n { {U - ,5) - 4 (1 + -i) 7/ - 2 (1 + 3, J) f ~ i,ii/ - ,i//' | = . (28)
Theorems I and II apply to (28) also. Then assume
it
.V being a positive integer. The equation for the determination of i/i is
2m- — wi = ; (20)
whence m — <jr -|- ^. Also .v ^ 1. The two integrals are expressible in
series form. Moreover the relations connecting the ^'s are, for Ijuth values
of 911, such that each coefficient is given as a multiple of the first one. The
two independent integrals of (28) may then be written
Z/i
1 -I- 1\. o.,j/
(30)
Vi = y* +■ -,■ ily'"^^ :
the «'s and ,9's being known.
i|
Ml
(:u)
112 I.INd. ON I'HK HOLimON OF A CKUTAIN Uin'KltKN I'lAl. KlH'-VTION
The coiupleto iiitogml of (28), tluni, is
n = 6-,y, -\- t'iy-i .
If now tho Hubstitutiou (27) bo lumlo in (1), an o(,uiition results which .lilVoM
from (2S) only by tho ox,.rossiou - K,^ (I ! yf i.. tho right h.iml uuMuber.
Of tliiH o.inution, tho eomplomontai-y function in givou by (:U). To con.p.ete
its intcn-ution, it Ih necessary to tiu.l a particular intogml. The character of
y .ui.rof th.' absolute term - h]i (1 + l/f make it clear thai a particular
i.ltegral can be obtaine.l in tin* form of a series of positive entire povyers.
Let this intc-ral b.^ .lenot.-.l bv ): It is not of iniportaiun* that its coethcionts
be calculate.1. TIum., in the neighborhood of .-• = 1, or ..f >J - 0, tho integral
of (28) cau bo exprosseil in the form
^\'J\ +
Then
Also
(la
(IVl ^ ,,, "yi .,. "
dy.,
<iy
dV
dy •
where / = I — 1 •
da dii .. —
•■• dl) ^ dx ' ^ '
dy ' ' <ly
■■= I -(z/- + 2//) = ;2/4r2'n/,
:-? ^ .,/^4 (2 + z/)5 j!j + 'vy^ ri + y)' %
+ 'y(2 + y)^'
■'y
(32)
When // - 0, the tirst and thinl terms on the right vanish, and the miiiaio
term becomes .
{du/dO)e.nyi - 6v7i 2 . (33)
In order, then, that at th.> 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
// <t, tlmt',, (» anil ('/«/'/")« T /a 0.
'I'liiis it iH Hccii that tlio cDuditioii stilted by Lord Kelvin roiinin's that
// = and II ---■ V. ConHoqnontly the series us written l»y Laplaee is the
complete solution for the eas(! of an earth completely covered to u constant
depth hy water.
Note. — Eipiation (Jj;}) {^ives an inniginary value iox {ilii/tl»)„, ^^.^ when thi'
arbitrary constant i\. is taken real. It is evident, however, that, when real
values of the function between .r, ■- U and x — 1 are desired, i:^ must he taken
purely imaginary ; for it was assumed that
1 + y,
(27)
KG that when x is h>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/</x it follows that
(iu/(lx = (' -\ /Hi - -^rri ,
where ( ' and /> are finite (and not zero) for all values of x.
15«t ., , ,.
t/„/,/ii - ii,i/</j! (1 - jr)'- - ^/(i - *-)' H Jf;
.: {dn/dfl)e=„,, = 1 ^(1 - .«-)■! -f- />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 <l I'll 'iiiij liro piii'dllrls of /iif.'/infe /i/lin/ in
one /ieiiii,y,,'ieri'.
5. Case of a c,ii.,tl lijiioj nlomj n /hir,i//,i of hititiole.
Cask 2.
1'.». /'o/iir xeo. If tln> 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 - •'.■:)^ <f'\ + ^^ £ [(1 -~ ^')' ">] = '> •
It has beea seen that
4- [(1 - •^")' "I'l i^ ^ *i"it<i quantity, say f> ;
-J- 1(1 — *'■)= v'l '** '^ ^^"'''" <l«'"itity, say f' .
(In one particular ciiso it is possible that a niiglit he Z'^ro.) Then it follows
that ^,,^^
Aa + ni> - . (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 <f'{u,) + " c («,) + ?/,'(«,) log «i + ".("i) ^- "i("i)log«i I
L "\ "\ 'h J
+ ii I ".' (",) -I 'f ". («,) 1 + V'(",) + \ V («,) = ,
f 9 1 '^ 1
^I I f'K) + " V'K) i "i'K)log«, + M,(//j + "i('A)log«2 I
H43)
As before.
A \s^ (.'•) I- n, {x) log ,r] H liu, ix) + V (.'•) - « = . (22)
Then, eliminating .4 and II, tiie equation for « is
V'W -f "iv'-)log.r ,•",(.'■) , V(^0
2 12 9
f 'K) -I f («i) + »<i'(«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('A!) log «:!. "i'(«i) I , «i(«2)
fC«,)
1
f '(«i) -i- "i('^i)
, V(«,)
, V'(«,)
,(44A)
9 a 1 2 '/ 2 2
f '(«2!* -t- " f K) -f '«i'(«j) log ' + "i(«2) + "if'-fo) lo^' -,«,'(«.) + ",{«o), i"(«,) + I'KO
//., //j '/.» '/.i '/] '/■■) (X.t
A siii..lar equatiou may be obtaiuoil for the evaluation of ii{a,).
Case 5.
22. ('(inul iif ir'xjth if] hj'nnj (ilomj <i, pii'dlli'.l of htt'itiil e. Suppose tlio
zonal sea to narrow down to a canal of width %l having as its northern bound-
ar\' the jiarallcl of latitude \z — ".
All tliriH' of the functions in the expression (22) are expressible in the
neighborhood of '/ =■■ H, in series form. Let them bo expressed in this man-
ner and let '/ be taken so small that powers of it higlier tlian the first may for
pur|)ost's of calculation be neglected.
TIh! second of the ecpxations (i;]) then becomes, after simplification by
means of the first one,
r 2 9 2 1
yi|f"(«i)-r V''<'«i) — "".. V'(«i) -t- , "i'('^i) ■- ,ih{a,)
[_ <l\ 'l\ "s "■\
2 2 1
+ ["i"(«i) H ".'(«i) - 7,»<i(«i)llog'/i
'/] "i J
L
J
+ d {I A + VI li + n) I 1 — a} = , (4{i)
where I, m, n may be easily found in terms of «,.
,li
WHICH I'KEHENTH ITSELF IN LAPLACE'h
?§
+
(>»
<M s"
+
C^l • r
KINETIC THEOIIV OF TIDES. 121
3-1
+
n-i
t^
■*
•JT ^ cc 5" >— 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 </<) beinj,' neglected the result is
2 7
f'(«i) + „ f('/) + -«,(«,)
3 ,
> ".'(«.) + I ".(«■) , V'(«,) + - V(«,)
•'i "■{ "■\ a, a.
f(«i)
, «,(«,)
^'(«.)
^ — Wi log «,
?»,
f'(«.) + - «l(«l)
. '</(«.)
, ^^"(«.)
+ S I 1 - «? ^'(«.) + ,"[ ",(«,) + ^ ^(«,)
",'('^.) -f I '«>(«.) , ^"(«.) + ~ i'(«,)
'*1 "l '*! Ui li^
V'(«l) + I f («l) + ■"■- M,(«,)
«1 «!
. "l'(«l) + „ "l(«l)
"1 '*1 «i «i
+ u I, 1 ~a^-<l
I — 111 log '/,
It is easy to show that
(48)
III,
I - III log a, = ,f-{a,) + ^."(«.) - ,;. f '(«i) + A f («,)
'*l '*l f^i
"i "■] '-i
2 4 4
m = «i"'('/,) + ■ *«,"('/,) — „ >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'^^^'
+
<r'(«,) ^ "/«■)
(49)
.He value o. . at th. .lie o. th. la.al . oUaiue^ h, puttin. . = ., ^^-^eh
U.e .idth n.ay be neglected (40) . ^^l^^^;^ ;, Uo aimensions.
. coiuc•iae^vith those obtained by cousiueiing the
In this fase the equation for u is
^■K) + ! f K) ' «. "'^"'^
«.'(«.) -r ,7 "■("')
"l ' ,.x u/., \ 1
, «,'(«,). S^"(«.)
(r>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])-
<rties of the integral important iu the applicatiou of the physical conditions
are deduciid.
20. S„ww,ini of V- VII. Section V deals with the ai)p]icatious of the
physical conditions to the determination of the arbitrary constants. One of
the constants is immediately determined. The determination of the other
involves greater diflicultie.s. The analysis on which i)revious evaluations have
i)een based is rej<'cted and replaced by a determination of the value which
appears to be entirely satisfactory. The condition which determines this
constant is the condition stated by Lord Kelvin. The result of this section
a[)pears to be a complete justification of Laplace's series.
WHICH I'BEHENTH ITSELF IN LAI'LACE'h KINETIC THEOUY OK TIDEH.
I'ir.
In Section VI tlio objoetiouH tiikou to tlio ])roof proviouHly f^ivon in thn
(It'tormiimtiou of tlio Hocond arl)itniry oonHtaiit iiic not forth.
The hiHt Sc'tion contniuH ilit^ diHCUHsion of five iniportimt cases in the
tli(iory of tides. The arl)itiaiy cojistants in the (general inte^'ial of tiu! dirt'er-
ential equation are deteiniined so that the intemids lepiesent tiie ti(hd distnrli-
nnce in those cnses, and expressions are obtained for tlie tidal disturbance at
any point whatever, and at certain particular points, snch as points on the
boundary. Tiie last of the five cases treated is that of a canal lyinfi; alonj^ a
])arallel of latitu<le and would apjcar to furiilHli a means of checking tlie same
case treated by Airy's Canal Theoiy o( Tides.
ReFERENC'FS TO LlTERATUnE OF THE PlIOltLFM.
I
Lai'LACE : Eecherches sur ))liisicur.i j)oints du Systeme du Monde ((Kuvr<!S,
t. ix).
Des oscillations de la nier et dc ratniosjdiere (Mecaniquo Celeste
Livre TV).
AiiiY : Tides and Waves (Encyclopedia Metropolitana).
On a controvcM-ted point in Laplac(!'s tiicory of tides ('Philosophical
Ma^'azine, ()ctoi)er, 1H75).
Kelvin: Note on an alleged error in Ija])lace's tlieory of tides (Philosophical
"*^af;azine, Septemlu^r, 187')).
Fekrel : Tidal llescarches (Ai.pi^ndix to tiie Ignited States Coast and Geodetic
Surv(!y l{(!port, 1S71).
On a controverted point in Laplace's tlujory of tides (Philosophical
jVEagazine, Marcii, l.S7(i, also (tould's Astntnomical Journal.
Vols. U and 10, and Smithsonian Miscellaneous Collections,
No. »-l'i).
Daiiwin : Tides (Encyclopedia JJritannica).
Basset : Treatise on Hydrodynamics, A'ol. II.
La.mb : Hydrodynamics, second edition.
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