De motu corponm circa punctum fixum mobilmm. 59 



^d® = — d|/3 cosg-+- t/rsin/) sin^, 

 d(Bd^ = {dp smq -+- dr sinp cosq) (dq — dr cosp) , 



^d^ = dq — drcosp, 



■^d^ = — dq-t-dr cosp, 

 d^d2 = {dpsinq -+- dr smp cosq) [dp cosq — dr sinp sing). 



70. Tum vero etiam habebitur ^; 



^d%-i-^d'^-+-(^d(B = 0, 



,^.,_^| M(^ -*-|5d^-f-3d3=0, ,^^ 



d[5t^-i- d3)^ -*- d®^ = dp^ -i- dr^ sla^ p = (dp cosq — dr sinp sing)^-»- (djo sing-i- (^r sin/) cosg)^, 

 dOS^H" d@^ H- d,§^ = (dp cosq — dr sinp sinqy-^- {dq — dr cosp)'^, 

 d(§.^-*-d^^ -H d^^ =(dqs\nq-\-drsinpcosqy-i-{dq — dr cospy. " " 



71. Cum igitur omnia ad has tres formulas 



dpcosq — (irsinpsing, d/) sing-i-drsin/) cosg et dq — drcosp 

 sint reducta , ponainus ad 'abbreviandum 



— dpcosq H- dr sinp sinq = Pdt, -i- dp slnq -h- dr slnp cosq = Qdt, — dq-^ dr cosp = Rdt, 



eruntque nostrae formulae . . 



^ .--,1.1) V... 



a(dq3H-2)d(EH-®d^=^-"Pcfe, ^d^-^^d^-+-(Bd^=-\-Qdt, ^d(^-h-^d^-^ ^d^ = — Rdt 

 33^31 -f-®d3) -*- ^d@ =;= -v-Pdt, ad5l -H ^d^-v- ^d(B= — Qdt, ©d93 -h gddn- 3^^ =-^Rdt 

 d%m-^dM(^-i-d(§d^=-QRdt'', dm^^-d^^d^^-i-d^^d^^-PRdl^ dm(^-i-d(§,d^-+-d^dS=-PQdt'- 



d^l^-i- d2)'-*- d^B^^dt"" {P^-i- Q^) 



^93=^ -f- d@2 _^_ t/^2 ^ ^^2 (p2 _^ /J2) 



72. Quod si jam hinc ad differentialia secunda descendamus, ob elementum temporis cfo con- 

 stans, reperiemus 



'nri')?.mj ■ '" toi 0J<? 



md^ -*- ^dd(g H- (Bdd^ = — dPfli -4- QRdt^ 

 ^dd% -\- (^dd^ H- ^dd® = -I- dPdt -i- j^Rdf^ 

 ^dd^ H- ^dd2) -f- 3dd® = — cZf3(if -+- PRdt"" 

 md^ -^^dd^^-^-^Bdd^ =-t-dQdt-t-PRdt^ 

 ^ ^dd(^ -t- (^dd^ -i- ^dd^ = — dRdt -t- PQdt^ 



«A<\U — " Qdd^-+- %dd(§, -*- ^dd^ = -t- dRdt -f- PQdf" 



Qldd5( -t- ^dd^ -+- (Bdd(B r= — df {PP -^ QQ) 

 ^dd^ -H (EdtZe -t- ^dd^ = — df2 (pp _|_ rr) 

 Qdd(B -H gddg -*- 3dd3 = — df^ {QQ -+- RR). 



\h 



