ROTATION" AND HORIZONTAL MOTIONS — BRASCHMANN 



point referred to the moving axes at the time t. Let (o t a> 2 co 3 be the 

 angular velocities of this point about these axes at the same instant. 

 Let oj be the resultant of these aagular velocities and let a, /?, y be 

 the coordinates of the origin of the movable system of axes with 

 reference to the fixed system; we have then the following expres- 

 sion for the accelerations: 



d 2 x, d 2 x „ / dz dy \ dw 2 du> 3 



— l . = — + 21 a/, - cu 3 — ) + z — - - y- — - 

 df- dt 2 \ dt dt dt dt 



+ io l {(0{X + oj 2 y + co 3 z) — or x + 



dt 2 ' 



d 2 y x _ d 2 y 

 dt- dt 2 



+ 2 



dx dz \ dm, daj. 



j 3 — co l t- v 



dt 



dt J 



dt 



dt 



d 2 d 

 + oj-, (oj. x + co^y + u)„z) — ory + 



dt 2 



d 2 z t d 2 z I dy dx \ 



— - = — + 2 oj. - — (o 2 — ) + y 

 dt 2 dt 2 \ dt ' dt 



■ (2) 



dcu. dco 9 



— x 



dt dt 



d 2 r 

 + oj 3 (w t x + w 2 y + co 3 z) - arz + — - ' 



dt 2 



[The detailed demonstration of these expressions was given by 

 Braschmann in the Bulletin of the St. Petersburg Academy for 

 185 1 and is repeated in Vol. I, Chap. IV of his Treatise on Theoret- 

 ical Mechanics and at pp. 74-88 of Erman's Archiv XXI. 1862; 

 its equivalent is found in many recent treatises on mechanics under 

 "constrained motion."] 



If the last named initial point or origin of the movable system 

 of coordinates be at the point on the earth's surface and movable 

 with it about the earth's axis then it has a constant velocity of rota- 

 tion about this axis. Hence the accelerations of its motion are 

 zero and therefore we have 



d 2 a 

 dt 2 



d'p 

 df 



*r 



dt 2 



= 



= f) 



(3) 



