[95] 



EULEE'S EQUATIONS OF MOTION. 



BY JAMBS LOUDON, M.A., 

 Professor of Mathematics and Natural Philosophy, University College, Toronto. 



1. A rigid body fixed at has at time t rotations (u^ m^ w^ round 

 the principal axes OA, OB, OC : to determine the changes per unit 

 time in these rotations. 



The positions OA', OB', OC of the axes at time t + 8t will be known 

 from the displacements in time 8t, due to these rotations, of the points 

 A (">!, o, o), ^(o, a>2) o), C{o, o, W3). The components of these displace- 

 ments in the directions OA, OB, OG, respectively, are evidently 

 o, m^w^dt, — (o^io^dt, for A ; 



^ (i)^co„8t, o, Ojio^dt, for B ', 



(o^w^di, — to^w^dt, o, for C . 



The component rotations at time t + dt are co^ + -j- -ot, &c., whicn 



may be represented by OA', OB', OC The changes of the rotations 

 in time 8t are therefore AA', BB' , GO'. E-esolving these changes into 

 the components {AF, FP, PA'), {BG, GQ, QB'), {GH, HR, RG'), in 

 the directions of the axes at time t, we get (observing that FP, PA' 



are the displacements in time 8t of the point F (to^ + -jr^^r °> °)j *^^-? 



and neglecting infinitesimals above the first order) the following as 



the resultant changes in time dt : 



, do)i . ^ dw, . , ^ . 



AF + GQ + HR = {-j^ - W2W3 + w^o}^) dt=~-dt along OA ; 



FP + BG + RG' = («'iW3 +-—- "^s"^!) St - -~-8t along OB ; 



PA' + QB'+ GH =.{- <o,(o^ + (o,co, - -^) H = -^^ dt along OG . 



du)^ dco^ d(o^ . , 

 The changes per unit tune are therefore —rr, "jT"? ~^' ^ 



directions OA, OB, OG, respectively. 



