42 



versor, regarded as the two principal /ae^o;"5 of any such qua- 

 ternion. 



II. To apply to problems of dynamics the foregoing 

 statical formulae, we have only to introduce, in conformity 

 with a well-known principle of mechanics, the consideration 

 of the equilibrium which must subsist between the forces lost 

 and gained. That is, we are to substitute for the symbol )3, 

 in the equations (1) or (2), the expression 



P = ^(^-S); (10) 



where m denotes the mass of that part or element of the sys- 

 tem which, at the time t, has a for its vector of position, and 



J2 



consequently -j-j for its vector of acceleration ;" while the new 



vector-symbol ^ denotes the accelerating force, or m<f) denotes 

 the moving force applied, direction as well as intensity being 

 attended to. Thus, instead of the two statical equations (2), 

 we have now the two following dynamical equations, for the 

 motion of a free but rigid system : 



S.m^ = S.m0; (11) 



^.mV.a^^=^.mV.a^; (12) 



of which the former contains the law of motion of the centre 

 of gravity, and the latter contains the law of the description 

 of areas. If the rigid system have one point fixed, we may 

 place at this point the origin of the vectors a; and in this case 

 the equation (II) disappears from the statement of the ques- 

 tion, but the equation (12) still remains : while the condition 

 that the various points of the system are to preserve unaltered 

 their distances from each other, and from the fixed point, is 

 expressed by the formula 



da ,- ,,„x 



^=r..a, (13) 



