## Sentence Examples

**The****process****of****transvection****is****connected****with****the****operations**12;**for**?**k**(**a****m****b****n**) = (**ab**)**kam-kbn-k**, (**x****y****x****y****or**2**S****k**(**a****x****by**)**x**= 4))**k**;**so****also****is****the****polar****process**,**for****since****f****k****m-k****k****k****n**-**k****k****y**=**a****x****by**, 4)**y**=**bx****by**,**if****we****take****the****k****th****transvectant****of****f****i****x**;**over**4**k**,**regarding****y**,,**y**2**as****the****variables**, (**f****k**, 4)**y**)**k**(**ab**)**ka****x**-**kb****k**(**f**, 15)**k**;**or****the****k****th****transvectant****of****the****k****th****polars**,**in****regard****to****y**,**is****equal****to****the****kth****transvectant****of****the****forms**.**Moreover**,**the****kth****transvectant**(**ab**)**k****a****m-k****b**: -**k****is****derivable****from****the****kth****polar****of****ax**,**viz**.**For****it****is****easy****to****establish**]**the****formula**(**yx**) 2 0 4 = 2f.4**-**2(**f****y**1) 2**connecting****the****Hessian****with****the****quartic****and****its****first****and****second****polars**;**now****a**,**a****root****of****f**,**is****also****a****root****of****Ox**,**and****con****se****uentl****the****first****polar**1**of****of****q****y****p****f**?**The****ellipsoids**(41)**and**(4~**i**)**are****reciprocal****polars****with****respect****to****a****sphere****having**0**as****centre**.**Trilinear****and****Tangential****Co-ordinates**.---**The****Geometrie****descriptive**,**by****Gaspard****Monge**,**was****written****in****the****year**1794**or**1 795 (7th**edition**,**Paris**, 1847),**and****in****it****we****have****stated**,**in****piano****with****regard****to****the****circle**,**and****in****three****dimensions****with****regard****to****a****surface****of****the****second****order**,**the****fundamental****theorem****of****reciprocal****polars**,**viz**.**And****iii**., 1810**-**1813);**and****from****the****theorem****we****have****the****method****of****reciprocal****polars****for****the****transformation****of****geometrical****theorems**,**used****already****by****Brianchon**(**in****the****memoir****above****referred****to**)**for****the****demonstration****of****the****theorem****called****by****his****name**,**and****in****a****similar****manner****by****various****writers****in****the****earlier****volumes****of****Gergonne**.**It****may****be****remarked****that****in****Poncelet'****s****memoir****on****reciprocal****polars**,**above****referred****to**,**we****have****the****theorem****that****the****number****of****tangents****from****a****point****to****a****curve****of****the****order****m**,**or****say****the****class****of****the****curve**,**is****in****general****and****at****most**=**m**(**m**- 1),**and****that****he****mentions****that****this****number****is****subject****to****reduction****when****the****curve****has****double****points****or****cusps**.**By****the****method****of****reciprocal****polars**)**deduce****from****it****the****other**,**but****we****do****at****one****and****the****same****time****demonstrate****the****two****theorems**;**our**(**x**,**y**,**z**.)**instead****of****meaning****point-co-ordinates****pay**,**mean****line-co-ordinates**,**and****the****demonstration****is****then****in****every****step****of****it****a****demonstration****of****the****correlative****theorem**.