## Sentence Examples

**Replacing****then****the****angle****i****on****the****right-hand****side****of****equations**(54) - (56)**by****some****mean****value**,**t**,**we****introduce****Siacci'****s****pseudovelocity****u****defined****by**(59)**u**=**q****sec**,**t**,**so****that****u****is****a****quasi-component****parallel****to****the****mean****direction****of****the****tangent**,**say****the****direction****of****the****chord****of****the****arc**.**The****equations**(66) - (71)**are****Siacci'**,**s****slightly****modified****by****General****Mayevski**;**and****now****in****the****numerical****applications****to****high****angle****fire****we****can****still****employ****the****ballistic****table****for****direct****fire**.**The****difficulty****is****avoided****by****the****use****of****Siacci'****s****altitude-function****A****or****A**(**u**),**by****which****y**/**x****can****be****calculated****without****introducing****sin****n****or****tan****n**,**but****in****which****n****occurs****only****in****the****form****cos****n****or****sec****n**,**which****varies****very****slowly****for****moderate****values****of****n**,**so****that****n****need****not****be****calculated****with****any****great****regard****for****accuracy**,**the****arithmetic****mean**1(0+0)**of**¢**and****B****being****near****enough****for****n****over****any****arc**4)-8**of****moderate****extent**.**Now****taking****equation**(72),**and****replacing****tan****B**,**as****a****variable****final****tangent****of****an****angle**,**by****tan****i****or****dyldx**, (75)**tan**4) -**dam**=**C****sec****n**[**I**(**U**) -**I**(**u**)],**and****integrating****with****respect****to****x****over****the****arc****considered**, (76)**x****tan**4, -**y**=**C****sec****n**(**U**) -**f**:**I**(**u**)**dx**] 0**But****f**(**u**)**dx**=**f**1(**u**)**du**=**C****cos****n****f****x****I**(**u**)**u****du****g****f**() =**C****cos****n**[**A**(**U**) -**A**(**u**)]**in****Siacci'****s****notation**;**so****that****the****altitude-function****A****must****be****calculated****by****summation****from****the****finite****difference****AA**,**where**(78)**AA**=**I**(**u**) 9 =**I**(**u**)**or****else****by****an****integration****when****it****is****legitimate****to****assume****that****f**(**v**) =**v****m****lk****in****an****interval****of****velocity****in****which****m****may****be****supposed****constant**.**In****the****application****of****Siacci'****s****method****to****the****calculation****of****a****trajectory****in****high****angle****fire****by****successive****arcs****of****small****curvature**,**starting****at****the****beginning****of****an****arc****at****an****angle**4)**with****velocity**v4),**the****curvature****of****the****arc**4**-**8**is****first****settled****upon**,**and****now**(80)**n**=1(0+0)**is****a****good****first****approximation****for****n**.**Institution**, 1888,**employing****Siacci'****s****method****and****about****twenty****arcs**;**and****Captain****Ingalls**,**by****assuming****a****mean****tenuity-factor****T**=0.68,**corresponding****to****a****height****of****about**2**m**.,**on****the****estimate****that****the****shot****would****reach****a****height****of**3**m**.,**was****able****to****obtain****a****very****accurate****result**,**working****in****two****arcs****over****the****whole****trajectory**,**up****to****the****vertex****and****down****again**(**Ingalls**,**Handbook****of****Ballistic****Problems**).**Siacci'****s****altitude-function****is****useful****in****direct****fire**,**for****giving****immediately****the****angle****of****elevation**4,**required****for****a****given****range****of****R****yds**.