2o'2 NOTES ON RELATIVE MOTION. 



At time t the whole moment of momentum is (employing OA, OB 



in a new sense) 



— /3w = OA along OS, 



— aw z= OB along Orj, 

 Go .... along OC, 



wliere a = 2my]!^, G = Sm [^ -j- rf"), etc. 



At time t -\- 8t the moment of momentum becomes 



— /5 (« + mdt) = OA' along 0|', 



— a (u) -\- wdt) = 0^' along Or]', etc. 



Hence the changes per unit time of moment of momentum are ulti- 

 mately -^) -j—i Gi), the components of which are — /3ti* + om"^ along 



0^, — au) — ftux^ along Orj, and CCj along OC. 



These, it will be observed, are of the same form as when the axes 

 are fixed in space. 



3. To measure the absolute velocity and acceleration of a point 

 referred to axes moving in space round 0. 



Let the motion of the axes be due to rotations (9j, Q.^, 0^ measured 

 along themselves. Then, proceeding as in § 1, the displacements of 

 a point P (^, Tj, Q due to these rotations are (C(?2 — 'J^s) ^^ along 0£, 

 (^03 _ ^d^) 8t along Of], and (vjf^i — ^d^)- 8t along OZ. These added to 

 the relative displacements {^dt, i]8t, ^di) of the moving point give 

 the absolute displacements. Hence the components of the absolute 

 velocity are 



u = OA = ^ -)- C02 — V^3 along 0^, 

 v=OB = rj -^-^$3 — C01 along Or;, 

 «; = OC = C + ^9i — ^^2 along OC 



Again, let the velocities at time t -\- dthe OA' = u -\- iidt along 



A A' RR' PC" 

 0^, etc.; then the absolute accelerations are ultimately —^t -j-> -r- ) 



whose components are 



ii — v'?3 + wO., along 01, 



V — W01 -|~ **^3 along Otj, 

 w — ud^ + ■^^i along OC. 

 These become, on reduction, 



1—2 d,n + 2 e^: + Zd,—y}h-{Qr + 01 + 01) ^ + (^^1 + rid, + CB,) 6, 



along 0^, etc. 



Note.— These resolutions are most readily eflfected as follows : AA' is 

 equivalent to ^D along Oij, DH along OK, and HA' along 0?; and similar 



