OF PABALLEL FLOTATIONS. 05 



and these give the motion of translation when the point (%' , y' , z') 

 is assumed for origin From these equations we have 



v! w„ -f- v' u>. , -f w' w_ = u w„ + v w„ + 10 w_. 



& y & w 2/ ~ 



Hence, if the velocity of translation be resolved into two, respec- 

 tively perpendicular to and along the axis of rotation, the latter re- 

 mains invariable in magnitude whatever origin be adopted ; and hence 

 also the velocity of translation will be the least possible when the 

 origin is so assumed that the former vanishes, or, iu other words, 

 when the velocity of translation is in the direction of the axis of 

 Rotation. 



If we seek an origin which shall make the motion of translation 

 vanish, or which shall make the whole motion reducible to a single 

 rotation, we must have for the determination of this origin (x' ,y' , z' ) 

 u' = o, v' = o,w' = o, 



o = u + «y' — <* z y 



o == v + <o s x' — (o x z' \. (i) 



O = 10 + OiJ/' — WyX' 



\ 



These equations are inconsistent unless a certain condition hold, 

 which is 



o = u w„ + v a> + wo> (2) 



When this condition is satisfied (provided w x , a> , ft>„ do not all 

 vanish), the equations (1) are equivalent to only two independent 

 equations and represent a straight line, every point of which is an 

 origin such as required. 



In the particular case where the motion consists of rotations round 

 parallel axes, taking u> as the type of one of these about an axis 

 through the point (x, y, z), and I, m, n for the direction-cosines of 

 the common direction of their axes, we have 



co e = I %(oi) ; o) = m S(ft)) ; ft)^ = n 2(&))- 



Also the linear velocity along the axis of x generated in the origin 

 of co-ordinates by one of these rotations o being n w y — m <o a 



we have 



u = n %(wy) — m %(<.»z) 



v = I 2)(ft)s) — n ~2<(iox) 



iv == m %(<tix) — I S(ft)y) 



The condition (2) is in this case satisfied, and (provided 5(w) do. 

 not vanish) the equations (1) assume the form 



, %(wx) f S(ft)J/) , %(a>z) 



2(a)) _ y S(a>) 2(a>) 



I 



