## Sentence Examples

**He****published****eight****folio****volumes****of****Greenwich****Observations**,**translated****Laplace'****s****Systeme****du****monde**(**in**2**vols**.**As****notable****instances****may****be****mentioned****Laplace'****s****discoveries****relating****to****the****velocity****of****sound****and****the****secular****acceleration****of****the****moon**,**both****of****which****were****led****close****up****to****by****Lagrange'****s****analytical****demonstrations**.**The****year**1787**was****rendered****further****memorable****by****Laplace'****s****announcement****on****the**19th**of****November**(**Memoirs**, 1786),**of****the****dependence****of****lunar****acceleration****upon****the****secular****changes****in****the****eccentricity****of****the****earth'****s****orbit**.**With****these****brilliant****performances****the****first****period****of****Laplace'****s****scientific****career****may****be****said****to****have****closed**.**To****this****task****the****second****period****of****Laplace'****s****activity****was****devoted**.- 4to,
**Paris**, 1799)**contains****methods****for****calculating****the****movements****of****translation****and****rotation****of****the****heavenly****bodies**,**for****determining****their****figures**,**and****resolving****tidal****problems**;**the****second**,**especially****dedicated****to****the****improvement****of****tables**,**exhibits****in****the****third****and****fourth****volumes**(1802**and**1805)**the****application****of****these****formulae**;**while****a****fifth****volume**,**published****in****three****instalments**, 1823**-**1825,**comprises****the****results****of****Laplace'****s****latest****researches**,**together****with****a****valuable****history****of****progress****in****each****separate****branch****of****his****subject**. **Expressions****occur****in****Laplace'****s****private****letters****inconsistent**3**Mee**.**This****instance****of****abnegation****is****the****more****worthy****of****record****that****it****formed****a****marked****exception****to****Laplace'****s****usual****course**.**These****researches****derive****additional****importance****from****having****introduced****two****powerful****engines****of****analysis****for****the****treatment****of****physical****problems**,**Laplace'****s****coefficients****and****the****potential****function**.**The****expressions****designated****by****Dr****Whewell**,**Laplace'****s****coefficients**(**see****Spherical****Harmonics**)**were****definitely****introduced****in****the****memoir****of**1785**on****attractions****above****referred****to**.**An****enumeration****of****Laplace'****s****memoirs****and****papers**(**about****one****hundred****in****number**)**is****rendered****superfluous****by****their****embodiment****in****his****principal****works**.**Laplace'****s****first****separate****work**,**Theorie****du****mouvement****et****de****la****figure****elliptique****des****planetes**(1784),**was****published****at****the****expense****of****President****Bochard****de****Saron**.**Some****of****Laplace'****s****results****in****the****theory****of****probabilities****are****simplified****in**.**S****Laplace'****s****treatise****on****specific****heat****was****published****in****German****in**1892**as****No**.**Sary****to****satisfy****Laplace'****s****equation****is****also****one****which****makes****the****potential****energy****a****minimum****and****therefore****the****energy****stable**.- 1875);
**Examples****of****Analytical****Geometry****of****Three****Dimensions**(1858, 3rd**ed**., 1873);**Mechanics**(1867),**History****of****the****Mathematical****Theory****of****Probability****from****the****Time****of****Pascal****to****that****of****Lagrange**(1865);**Researches****in****the****Calculus****of****Variations**(1871);**History****of****the****Mathematical****Theories****of****Attraction****and****Figure****of****the****Earth****from****Newton****to****Laplace**(1873);**Elementary****Treatise****on****Laplace'**,**s****Lame'****s****and****Bessel'****s****Functions**(1875);**Natural****Philosophy****for****Beginners**(1877). **That****is****to****say**,**instead****of****using****Boyle'****s****law**,**which****supposes****that****the****pressure****changes****so****exceedingly****slowly****that****conduction****keeps****the****temperature****constant**,**we****must****use****the****adiabatic****relation****p**=**kpy**,**whence****d****p**/**d****p**=**y****k****p****Y**1=**yp**/**p**,**and****U**= (**yp**/**p**) [**Laplace'****s****formula**].**In****the****first****of****these**,**entitled**"**Recherches****sur****l'attraction****des****spheroides****homogenes**,"**published****in****the****Memoires****of****the****Academy****for**1785,**but****communicated****to****it****at****an****earlier****period**,**Legendre****introduces****the****celebrated****expressions****which**,**though****frequently****called****Laplace'****s****coefficients**,**are****more****correctly****named****after****Legendre**.**The****introduction****of****the****coefficients****now****called****Laplace'**,**s****and****their****application**,**commence****a****new****era****in****mathematical****physics**."**The****third****memoir****relates****to****Laplace'****s****theorem****respecting****confocal****ellipsoids**.**For****the****first****time****we****have****a****correct****and****convenient****expression****for****Laplace'****s****nth****coefficient**."**His****well-known****correction****of****Laplace'****s****partial****differential****equation****for****the****potential****was****first****published****in****the****Bulletin****de****la****societe****philomatique**(1813).**In****the****theory****of****surfaces**,**in****hydrokinetics**,**heat-conduction**,**potentials**, &**c**.,**we****constantly****meet****with****what****is****called**"**Laplace'****s****operator**,"**viz**.**And****we****now****see****that****the****square****of****V****is****the****negative****of****Laplace'****s****operator**;**while****V****itself**,**when****applied****to****any****numerical****quantity****conceived****as****having****a****definite****value****at****each****point****of****space**,**gives****the****direction****and****the****rate****of****most****rapid****change****of****that****quantity**.**Leslie'****s****theory****was****afterwards****treated****according****to****Laplace'****s****mathematical****methods****by****James****Ivory****in****the****article****on****capillary****action**,**under**"**Fluids**,**Elevation****of**,"**in****the****supplement****to****the****fourth****edition****of****the****Encyclopaedia****Britannica**,**published****in**1819.**He****proceeded****to****an****investigation****of****the****equilibrium****of****a****fluid****on****the****hypothesis****of****uniform****density**,**and****arrived****at****the****conclusion****that****on****this****hypothesis****none****of****the****observed****capillary****phenomena****would****take****place**,**and****that**,**therefore**,**Laplace'****s****theory**,**in****which****the****density****is****supposed****uniform**,**is****not****only****insufficient****but****erroneous**.**In****particular****he****maintained****that****the****constant****pressure****K**,**which****occurs****in****Laplace'****s****theory**,**and****which****on****that****theory****is****very****large**,**must****be****in****point****of****fact****very****small**,**but****the****equation****of****equilibrium****from****which****he****concluded****this****is****itself****defective**.**Hence****we****may****write****p**=**P**+Ap2,**where****A****is****a****constant**[**equal****to****Laplace'****s****intrinsic****pressure****K**.**The****expression****for****the****intrinsic****pressure****is****thus****simply****K**= 2**iro**2**f**1,**G**(**z**)**dz**(28)**In****Laplace'****s****investigation****o**-**is****supposed****to****be****unity**.**From****this****consideration****we****may****derive****Laplace'****s****expression**,**as****has****been****done****by****Dupre**(**Theorie****mecanique****de****la****chaleur**,**Paris**, 1869),**and****Kelvin**("**Capillary****Attraction**,"**Proc**.**Roy**.**Since**0(**o**)**is****finite**,**proportional****to****K**,**the****integrated****term****vanishes****at****both****limits**,**and****we****have****simply****f**0(**z**)**dz****f**: (**z**)**dz**, (34)**and****T**=**ref**: z1,1,(**z**)**dz**(35)**In****Laplace'****s****notation****the****second****member****of**(34),**multiplied****by**27r,**is****represented****by****H**.**For****further****calculations****on****Laplace'****s****principles**,**see****Rayleigh**,**Phil**.- ("
**Laplace'****s****Theory****of****Capillarity**,"**Rayleigh**,**Phil**. **Mag**., 1883,**P**. 315)**According****to****Laplace'****s****hypothesis****the****whole****energy****of****any****number****of****contiguous****strata****of****liquids****is****least****when****they****are****arranged****in****order****of****density**,**so****that****this****is****the****disposition****favoured****by****the****attractive****forces**.**So****far****the****results****of****Laplace'****s****hypothesis****are****in****marked****accordance****with****experiment**;**but****if****we****follow****it****out****further**,**discordances****begin****to****manifestthemselves**.**Laplace'****s****mathematical****theory****of****the****form****of****Saturn'****s****rings**.