76, 77] RELATIVE MOTION. 247 



Making use of this value of I with equations 121), 123), we obtain 

 for v for points on the central -axis 



125) V = 



2r 

 agreeing with 42). 



If the velocity of points on the central axis is to be zero, we 

 must have 



126) p v x + q.V y + rV, = Q, 

 when the motion reduces to a rotation, as in 41). 



77. Relative Motion. In forming equations 114) and the 

 following, we have supposed the point in question fixed in the body, 

 so that x, y, z were constants. If this is not the case we have to 

 add to the right hand members of 114) the quantities 



dx . a dy . dz 



^-st + ^^^dt' 



IOTN dx . a dy , dz 



127 ) *-& + &- + **' 



dx , dy , dz 



"di + hdi + v*di' 



which, on being multiplied by the proper cosines, will appear in 

 equations 117) as -^j _> -^> so that we have for the components 



of the actual velocity in the direction of the axes X, Y, Z at the 

 instant in question, if the origin of the latter is fixed, 



dx , 

 v *Tjf + &"~ ry > 



128) ^g+-^ 



dz 



These equations are of very great importance, for by means of them 

 we may express the velocity components in directions coinciding 

 with the instantaneous direction of the moving axes of the end of 

 any vector x, y, z. If for x, y, 2 we put the components of the 

 velocity v, we obtain the acceleration -components ( 103), if the 

 components of angular momentum H we have a dynamical result 

 treated in 84. 



