De motu corporum in tuho mobih circa axem fixum. 79 



Hi valores ipsarum m et />, in aequatione d$ = -y — substituti, dabunt aequationem 



, {Aaa-+- Mkk)dxVc -,//» c {Aaa-t- Mkk) {xx — aa)'. 



Axx -+- Mkk * ''^^ ~* ff{Axx-i-Mkk) ' * 



qua natura curvae ^^P, quam corpus revera describit, determinatur. 



17. Ponamus corpus initio in J nullam habuisse celeritatem in tubo progressivam , seu esse 

 6 = 0, erit ds= . "" "^ — r^TTT' Hinc centro ducto arculo Pl. erit 



' V{xa; — aa)(Axx-i-Mkk) 



p, xds xdxV{Aaa -*- Mkk) 



f V {xx — aa) {Axx -t- Mkk) '. ' 



et pl = dXf mde flt elementum curvae ^ 



Pp = dxV(i-+- ^^M««-*-W) ^ ^ ^^ yAx^-^Mkk{^xx--aa) ^ 

 " ^ (a;a7 — aa) {Axx n- itfM) '^ {xx — aa) {Axx -f- Mkk) ^ 



Pp:Pl= y{Jx'' -H (2cca; — aa) Mkk) : xV{Jaa -+- Mkk). 

 Ipso ergo initio in J, erat Pp=Ply ideoque tangens curvae JP in ./^ erat normalis ad tubum OF. 

 )n P autem est cos JPO = ^ = y(^.-aa)(^.-.-^a ) ^_^ ^^ g^. 



1 Pp Ax* H- (2a;a; — aa)Mkk 



c\ jnr\ Ax* — '2Aaaxx — Mkkaa . 'iMkkxx — '2Aaaxx 



COS 2AP0 = —-. 7^ : — — = 1 



■jn' 



Axi^ -\- {^x — aa)Mkk Ax*^ -t- (2a;a; — aa)Mkk 



n: 18. Ponamus tubum omni inertia destitutum, ita ut sit Mkk = Oy eritque 



■y\i - '■.■ uit>J Uiti. a*c r aa ^ , 



u=-^j seu Vu= -Vc, 



•: - "■ ■ X* XX 



oi:,. • ^■■. :.:; ;. -. : 



unde celeritas tubi in F erit ad celeritatem tubi in 5" ut QP^ ad OA^. Deinde autem fiet 



, aac{xx — aa) ,. . , dxVu aafdxVc 



p = b-i -j unde ent ds = -^=—y— — — rr» 



' ffxx Vp xV{bffxx-^aacxx—a*c) 



quae aequatio integrata dat ^ ^^^^* ' 



,<\Vk idot mwoiJoMil) fpili.(. „„y, ^y^ 



_- = arc. cos -y- — — arc. cos — — = ang. FOS. 



;}'>??'» M : ;.r f xV{aac-t-bff) V{aac-t-bff) » 



At est arc.cos;/— ^-^-— =arc. tang^=90° — FJP,, ui^de 



. V {aac -t- bfff^) ° aVc . . -~ x 



<.!.?oi! V .!fiff.r,-,vjfvl'»ff arc. COS -77^^-^— = 90° -r- F.iP -H FOS" et 



' ■ . xV {aac -t- bff^) . , 



1- II . ,/ ,^ = cos (90^— FAP H- FQ5') = sin (FAP-^FOS) = - sin FAP, 



ob ,^ °^' ^^ = cos (90°— F^P\="sin F^P. 



V{aac-^bff^ ^ ' 



Hinc erit x:a = OP.OA= sin F^P : sin {FAP — FOS)y quo constat viam a corpore descriptam 

 AP esse lineam rcctam; tum ^im erit 



: r^ ,.fu, ^„»|.o} . F^P — FOS = APO et OP:OA = sin FAP : sin ^PO. 

 Ex ipsa autem status natura perspicuum cst, corpus perinde motum iri, ac si tubus penitus abesset, 

 quia inertia carens motum corporis alterare ncquit. 



i-a 



