## 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**.**On****the****right-hand****side****is****such****that****the****coefficient****of****x****n****ix****n****Zx****n**3...**If****therefore****we****choose****a****quantity****e****such****that****log****e****I****o****X****X**=**I**,**log****i****oe**=**X**,**which****gives**(**by****more****accurate****calculation**)**e**=2.71828...,**we****shall****have****lim**(**loge**(**I**+0))}/0=**I**,**and****conversely**'**lim**' {**ex**+0 -**e****x**} 143=**The****deduction****of****the****expansions****log****e**(' +**x**) =**x**-**Zx**2 + 3x 3 - ...,**e****x**=**I**+.**x**+x2/2!+x3/3!-}-...,**is****then****more****simply****obtained****by****the****differential****calculus****than****by****ordinary****algebraic****methods**.**When****the****plane****zx****is****not****a****plane****of****symmetry**,**we****have****to****consider****the****terms****in****xy**, 2**y**,**and****y**3 .**Kirchhoff'****s****expressions****for****X**,**Y**,**Z**,**the****coordinates****of****the****centre****of****the****body**,**FX**=**y**1**cos****xY--y**2**cos****yY**-{-**y**3**cos****zY**, (18)**FY**= -**y****l****cos****xX**-Hy2**cos****yX**+**y**3**cos****zX**, (**Ig**)**G**=**y**1**cos****xZ**+**y**2**cos****yZ**+**y**3**cos****zZ**, (20) (21)**F**(**X**+**Yi**) = Fy3**-**Gx3+**i**/)**X**3epi.**Since****loge**(**I**+**x**) =**x-**2x 2 -3x 3 - 4x4+&**c**.,**we****have**,**by****changing****the****sign****of****x**,**log****e**(**I**-**x**) _ -**x**-**zx**2 - 3x 3 -**x**4 - &**c**.;**whence****g**1 +**x****to**=2(**x**+**ix**'+1x5+&**c**.),**e****l**-**x****and**,**therefore**,**replacing****x****by****p**+**q**,**log****e****q**=2**p**+**q**+3 () 3T ?**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**').**To****find****the****pressure****exerted****by****a****bar****AB****on****the****pin****A****we****compound****with****the****force****in****AB****given****by****the****diagram****a****force****equal****to****P**.**Conversely**,**to****find****the****pressure****of****the****pin****A****on****the****bar****AB****we****must****compound****with****the****force****given****by****the****diagram****a****force****equal****and****opposite****to****P**.**This****question****arises****in****practice****in****the****theory****of****three-jointed****structures**;**for****the****purpose****in****hand****such****a****structure****is****sufficiently****represented****by****two****bars****AB**,**BC**.**The****right-hand****figure****represents****a****portion****of****the****force-diagram**;**in****particular****ZX****represents****the****pressure****of****AB****on****B**