## Sentence Examples

**But****taking**,**for****example**,**Simpson'****s****mean****values****at****Karasjok**,**we****have****observed****p**-**I**+ -**I****i**= 0.05**X**(**cm**.**This****involves****the****use****of****Cartesian****co-ordinates**,**and****leads****to****important****general****formulae**,**such****as****Simpson'****s****formula**.**The****application****of****Simpson'****s****rule**,**for****instance**,**to****a****plane****figure****implies****certain****assumptions****as****to****the****nature****of****the****bounding****curve**.**For****this****reason**,**formulae****which****will****only****give****approximate****results****are****usually****classed****together****as****rules**,**whether****the****inaccuracy****lies**(**as****in****the****case****of****Huygens'****s****rule**)**in****the****formula****itself**,**or**(**as****in****the****case****of****Simpson'****s****rule**)**in****its****application****to****the****data**.**This****is****Simpson'****s****formula**.**This****is****Simpson'****s****second****formula**.**Simpson'****s****two****formulae****also****apply****if****u****is****of****the****form****px**3 - }- 5x 2 +**rx**-}-.**s****Simpson'**(**s****first**)**formula**,**for****instance**,**holds****for****f**=**I**,**and****is****obtained****by****taking****p**=**i****and****ignoring****differences****after**52up.**Thus****we****find****from**(**i**)**that****Simpson'****s****second****formula**,**for****the****case****where****the****top****is****a****parabola**(**with****axis**,**as****before**,**at****right****angles****to****the****base**)**and****there****are****three****strips****of****breadth****h**,**may****be****replaced****by****area**= 8h(3u**i**+ 2U 1 + 3us).**This****might****have****been****deduced****directly****from****Simpson'****s****first****formula**,**by****a****series****of****eliminations**.**It****follows****from**§§ 48**and**51**that**,**if****V****is****a****solid****figure****extending****from****a****plane****K****to****a****parallel****plane****L**,**and****if****the****area****of****every****cross-section****parallel****to****these****planes****is****a****quadratic****function****of****the****distance****of****the****section****from****a****fixed****plane****parallel****to****them**,**Simpson'****s****formula****may****be****applied****to****find****the****volume****of****the****solid**.**The****areas****of****the****sides****for****which**0**and****x**=**xo**+2h,**and****of****the****section****by****the****plane****x**=**xo**+**h**,**may****be****found****by****Simpson'****s****second****formula**;**call****these****Ao****and**A2,**and****Al**.**Hence****Simpson'****s****formula****applies**,**and****the****volume****is****lh**(A0**-I-**4A1+ A2).**Applying****Simpson'****s****formula****to****each****of****these**,**we****have****A**-__9= ***h**(**uo**+ 4 14 1 + u2) + 3**h**(14 2 + 4u3 + u4) + ..- + 214m
**-**2 + 4**u****m-i**+**um**)**This****is****Simpson'****s****rule**. - (
**ii**)**Similarly**,**if****m****is****a****multiple****of**3,**the****repeated****application****of****Simpson'****s****second****formula****gives****Simpson'****s****second****rule****A**1? **Consider**,**for****example**,**Simpson'****s****rule**(§ 68 (**i**)).- +
**u****m-p**+**zum**),**which****may****be****denoted****by****Cp**.**With****this****notation**,**the****area****as****given****by****Simpson'****s****rule****may****be****written****in****the****form****sC****l**- 3 C2**or****CI**+ 1 3 `-(C1C2). - (
**i**)**m****a****multiple****of**2 (**Simpson'****s****rule**). - (
**ii**)**m****a****multiple****of**3 (**Simpson'****s****second****rule**). **The****formula****applied****can****then****be****either****Simpson'****s****rule****or****a****rule****based****on****Gauss'****s****theorem****for****two****ordinates**(§ 56).**These****theorems****may****prove****useful****in****preliminary****calculations****where****the****pressure-curve****is****nearly****straight**;**but**,**in****the****absence****of****any****observable****law**,**the****area****of****the****pressure-curve****must****be****read****off****by****a****planimeter**,**or****calculated****by****Simpson'****s****rule**,**as****an****indicator****diagram**.**Besides****these****public****establishments****for****the****custody****of****lunatics**,**there****are****in****the****vicinity****of****Dublin****various****private****asylums**.**The****principal****institution****for****blind****men**(**and****also****those****afflicted****by****gout**)**is****Simpson'****s****hospital**(1780),**founded****by****a****merchant****of****Dublin**;**while****blind****women****are****maintained****at****the****Molyneux****asylum**(1815).**An****admirable****biography****is****to****be****found****in****Richard****Simpson'****s****Edmund****Campion**(1867);**and****a****complete****list****of****his****works****in****De****Backer'****s****Bibliotheque****de****la****compagnie****de****Jesus**.