108 Proceedings of the Royal Irish Academy. 



Thus, for a rigid body we have the following equations of impulsive 

 motion (which are independent of the mutual actions of the parts of the 

 body) : — 



^»^lPl = 2fi, %nh V^pipi = % V^piii, and ^m^pi^ = '$Spi$i. 



The above detailed treatment has been given in order to detect, if 

 possible, quantities analogous to couples of higher orders which could 

 not occur in three dimensions. Por, if the equation mipi = ii is multi- 

 plied by products such as pip^, p^^, &c., it will be found impossible, on 

 summation, to get rid of the mutual impulses. 



45. If Pq is the vector to the centre of mass, 



Pi = Po + '^1, and Pi = Po + ^1 = Po + ^2^^i> 

 where O is a quadratic function of the units corresponding to angular 

 velocity, and already treated of in Art. 10. Hence, the dynamical 

 equations are, if M is the total mass, ^ the resultant impulsive force, 

 r the impulsive couple, and ^the kinetic energy, 



Mp, = i, Mr,p,p, + %mV,z:V,nz=y = T, 



and Mpo^ + ^m ( V^nzjf = -2T; 



or, if %m Fs^ F^Oct = $0, 



Jfpo = i, M Vop,p, + <l>0 = r, and Mp,^ + SQ^n = - 2 J. 



In these, <J>Q is a linear function of O quadratic in the units. 

 Observing that O may be expressed in the form ^V^Xp., where 

 A and /x are linear vectors, 



$0 = S* VoX/x = % . ^m FaW F; ViXfJL . ^. 



Now, exactly as in Quaternions, 



F^ . F'oA./x . "J? = XSp,7s — fxSX'^, 

 and hence 



$0 = % Vzp. {'^mzjSXts) - 2 FjX (^mzsSp,'^) = %Vz iiJ-<i>X - A</)/x), 



if <^ is the linear vector function defined by <jSp = %mT^Spz:. This 

 function is self-conjugate, and its axes are consequently real and 

 mutually perpendicular. Let these be iii^ . . . «',„, and let 



■Z3- = i^Xi + ^2^o + &c. 



Then cfji^ = - %m {i^Xi + ioX^ -\- . . .) x^ = - i^mx^ = - g^i-^ (say), 



and %mXiX2 = &c. = ; 



so the units iii-^, &c., are parallel to the principal axes of inertia of the 

 body. 



