## Sentence Examples

**Xvi**.),**to****which****is****appended****an****appreciation****of****Plucker'****s****physical****researches****by****Hittorf**,**and****a****list****of****Pliicker'****s****works****by****F**.**Pliicker'****s****Equations**.**In****regard****to****the****ordinary****singularities**,**we****have****m**,**the****order**,**n**„**class**, „**number****of****double****points**,**Cusps**,**T****double****tangents**,**inflections**;**and****this****being****so**,**Pliicker'**”**s****six****equations**”**are****n**=**m**(**m**-**I**) -2S -3K, = 3m (**m**- 2) - 6S- 8K,**T**=**Zm**(**m**-2) (m29) - (m2 -**m-**6) (28**-i-**3K)-**I**-25(5**-**1) +65K**-**1114**I**),**m**=**n**(**n**-**I**)-2T**-**3c,**K**= 3n (**n-**2) - 6r -8c, = 2n(**n-**2)(n29) - (n2 -**n-**6) (2T-{-30**-**1**-**2T(**T**-**I**) -1**-**6Tc -}2c (**c**-**I**).**It****is****implied****in****Pliicker'****s****theorem****that**,**m**,**n**,**signifying****as****above****in****regard****to****any****curve**,**then****in****regard****to****the****reciprocal****curve**,**n**,**m**,**will****have****the****same****significations**,**viz**.**With****regard****to****the****demonstration****of****Pliicker'****s****equations****it****is****to****be****remarked****that****we****are****not****able****to****write****down****the****equation****in****point-co-ordinates****of****a****curve****of****the****order****m**,**having****the****given****numbers**6**and****of****nodes****and****cusps**.**We****have****thus****finally****an****expression****for**=**m**(**m-**2) (m2**-**9) - &**c**.;**or****dividing****the****whole****by**2,**we****have****the****expression****for****given****by****the****third****of****Pliicker'****s****equations**.**It****is****obvious****that****we****cannot****by****consideration****of****the****equation****u**=**o****in****point-co-ordinates****obtain****the****remaining****three****of****Pliicker'****s****equations**;**they****might****be****obtained****in****a****precisely****analogous****manner****by****means****of****the****equation****v**=**o****in****line-co-ordinates**,**but****they****follow****at****once****from****the****principle****of****duality**,**viz**.**To****complete****Pliicker'****s****theory****it****is****necessary****to****take****account****of****compound****singularities**;**it****might****be****possible**,**but****it****is****at****any****rate****difficult**,**to****effect****this****by****considering****the****curve****as****in****course****of****description****by****the****point****moving****along****the****rotating****line**;**and****it****seems****easier****to****consider****the****compound****singularity****as****arising****from****the****variation****of****an****actually****described****curve****with****ordinary****singularities**.**So****that**,**in****fact**,**Pliicker'****s****equations****properly****understood****apply****to****a****curve****with****any****singularities****whatever**.**By****means****of****Pliicker'****s****equations****we****may****form****a****table**-**The****table****is****arranged****according****to****the****value****of****in**;**and****we****have****m**=**o**,**n**=**r**,**the****point**;**m**=1,**n**=**o**,**the****line**;**m**=2,**n**=2,**the****conic**;**of****m**= 3,**the****cubic**,**there****are****three****cases**,**the****class****being**6, 4**or**3,**according****as****the****curve****is****without****singularities**,**or****as****it****has**1**node****or****r****cusp**;**and****so****of****m**=4,**the****quartic**,**there****are****ten****cases**,**where****observe****that****in****two****of****them****the****class****is**= 6, -**the****reduction****of****class****arising****from****two****cusps****or****else****from****three****nodes**.**At****any****one****of****the****m**2 -26 - 3K**points****the****variable****curve****and****the****consecutive****curve****have****tangents****distinct****from****yet****infinitesimally****near****to****each****other**,**and****each****of****these****two****tangents****is****also****infinitesimally****near****to****one****of****the****n**2 -2T**-**3t**common****tangents****of****the****two****curves**;**whence**,**attending****only****to****the****variable****curve**,**and****considering****the****consecutive****curve****as****coming****into****actual****coincidence****with****it**,**the****n**2 -2T**-**3c**common****tangents****are****the****tangents****to****the****variable****curve****at****the****m**2 -26**-**3K**points****respectively**,**and****the****envelope****is****at****the****same****time****generated****by****the****m**2 -26**-**3K**points**,**and****enveloped****by****the**n2**-**2T**-**3c**tangents**;**we****have****thus****a****dual****generation****of****the****envelope**,**which****only****differs****from****Pliicker'****s****dual****generation**,**in****that****in****place****of****a****single****point****and****tangent****we****have****the****group****of**m2**-**26**-**3K**points****and****n**2 -2T**-**3c**tangents**.**We****can****by****means****of****it****investigate****the****class****of****a****curve**,**number****of****inflections**, &**c**. -**in****fact**,**Pliicker'****s****equations**;**but****it****is****necessary****to****take****account****of****special****solutions**:**thus**,**in****one****of****the****most****simple****instances**,**in****finding****the****class****of****a****curve**,**the****cusps****present****themselves****as****special****solutions**.**The****system****has****singularities**,**and****there****exist****between****m**,**r**,**is****and****the****numbers****of****the****several****singularities****equations****analogous****to****Pliicker'****s****equations****for****a****plane****curve**.