## Sentence Examples

**If****three****equations**,**each****of****the****second****degree**,**in****three****variables****be****given**,**we****have****merely****to****eliminate****the****six****products****x**, 2,**z**2,**yz**,**zx**,**xy****from****the****six****equations****u**=**v**=**w**=**o**=**oy**= = 0;**if****we****apply****the****same****process**:**to****thesedz****equations****each****of****degree****three**,**we****obtain****similarly****a****determinant****of****order**21,**but****thereafter****the****process****fails**.**With****the****values****above****of****u**,**v**,**w**,**u**',**v**',**w**',**the****equations****become****of****the****form****p****x**+ 4 7rpAx -**Fax**-{-**hy**-}-**gz**=**o**, -**p**-**dy**+ 4?**pBy**+**hx**+**ay**+**fz**=**o**,**P****d****p**+**TpCZ**+**f****y**+**yz**=**o**,**and****integrating****p****p**1+27rp(Ax2+By2+CZ2) +**z**('**ax****e**+**ay****e**+ yz2 2**f****yz**+ 2gzx + 2**hx****y**) =**const**., (14)**so****that****the****surfaces****of****equal****pressure****are****similar****quadric****surfaces**,**which**,**symmetry****and****dynamical****considerations****show**,**must****be****coaxial****surfaces**;**and****f**,**g**,**h****vanish**,**as****follows****also****by****algebraical****reduction**;**and**4c2 (**c**2 - a2)?**Two****spheres****intersect****in****a****plane**,**and****the****equation****to****a****system****of****spheres****which****intersect****in****a****common****circle****is****x**2 +**y**2 +**z**2 +2Ax -**fD**=**o**,**in****which****A****varies****from****sphere****to****sphere**,**and****D****is****constant****for****all****the****spheres**,**the****plane****yz****being****the****plane****of****intersection**,**and****the****axis****of****x****the****line****of****centres**.**But**,**assuming****the****distributive****principle**,**the****product****of****two****lines****appeared****to****give****the****expression****xx**' -**yy**' -**zz**' +**i**(**yx**' +**xy**')+**j**(**xz**'**i****j**(**yz**' +**zy**').**But****the****numerical****factor****appears****to****be****yz**'+**zy**',**while****it****is****the****quantity****yz**' -**zy**'**which****really****vanishes**.**For****his****speculations****on****sets****had****already****familiarized****him****with****the****idea****that****multiplication****might****in****certain****cases****not****be****commutative**;**so****that**,**as****the****last****term****in****the****above****product****is****made****up****of****the****two****separate****terms****ijyz**'**and****jizy**',**the****term****would****vanish****of****itself****when****the****factorlines****are****coplanar****provided****ij**= -**ji**,**for****it****would****then****assume****the****form****ij**(**yz**' -**zy**').**He****had****now****the****following****expression****for****the****product****of****any****two****directed****lines**:**xx**' -**yy**-**zz**' +**i**(**yx**'+**xy**')+**j**(**xz**' '+**zx**') +**ij**(**yz**' -**zy**').**And****now****a****directed****line****in****space****came****to****be****represented****as****ix**+**jy**+**kz**,**while****the****product****of****two****lines****is****the****quaternion**- +**yy**' +2z') +**i**(**yz**' -**zy**') +**j**(**zx**' -**xz**') +**k**(**x****y**' -**yx**').**Also**,**if**0**be****the****angle****between****them**,**and****x**",**y**",**z**"**the****direction-cosines****of****a****line****perpendicular****to****each****of****them**,**we****have****xx**' +**yy**'+**zz**' =**cos**0,**yz**' -**zy**" =**x**"**sin**0, &**c**.,**so****that****the****product****of****two****unit****lines****is****now****expressed****as**- cos0+ (**ix**" +**jy**" +**kz**")**sin**0.**Thus****the****sum**~(**m**.**yz**)**is****called****the****product****of****inertia****with****respect****to****the****planes****y**=**o**,**z**=**o**.**Yz**) =~(**m**.,**l**)+**Z**(**m**).**Now****if****the****displacement****z****be****everywhere****very****small**,**the****curvature****in****the****planes****parallel****to****xz****and****yz****will****be****d**2**z**/**dx**2**and****d**2**z**/**dy****e****respectively**,**and****if****T****is****the****surface-tension****the****whole****upward****force****will****be****d**2**z**d2zl**T**(4x 2 + + (**p**-**o**)**gz**.