118 Proceedings of the Royal Irish Academy. 



If the new quadratic C is defined by 



12 = 1 h &c., 



e = V^Q.' (cTo - cr) + ezm+ihm+ii 



in wliicli eom+i is arbitrary; and when the base-point is transferred to 

 the extremity of this vector (a definite poifit in even space, because 

 hm+i cannot occur), the velocity of translation (cr) is reduced to 



an actual and real vector in odd spaces, but zero in even spaces. In 

 other words, a body moving freely and with perfect generality in an 

 odd space instantaneously contains a line whose position is deter- 

 mined, and the particles of the body situate on that line are moving 

 along it with a determined velocity ; but a body moving with equal 

 freedom in an even space has one point instantaneously fixed, and the 

 position of that point is determined. 



56. The difiiculty in dealing with the equation cr = (rQ+ViQe arises 

 solely from the fact that Q,~^ is not of the same character as O, viz. : — 

 a quadratic in the units ; and the reason of this is, the area vectors 

 {hh)> (Vi)) &c., in Q are commiitative, and not polar in multiplication. 



However, multiplying by O"^, 



because fie = {Vi+ V^) fie. 



Before going farther, it is necessary to consider the nature of the 

 quPvutity fi"^ Fsfie. I am not yet able to give a satisfactory account 

 of this quantity, as I have not worked out fully the nature of fi"\ 

 It is not hard to see that fi"^ must be proportional to the product 

 obtained by multiplying together the results of changing the signs of 

 the area vectors in fi in all possible ways. 



Again, Fsficr is not independent of the base-point, nor do I see how 

 to operate on cr = o-g + Fjfie, so as to obtain a result independent of e, 

 or to obtain an expression analogous to the pitch of a wrench. 



57. I shall now consider a special class of operators -P ( ) -P"\ which 

 permute the units iii^ • > .in among themselves. 



In Ai't. 36, let 



ji = h, j\ = iz, &c. ; y,„_i = 4, and j„, = ± i^. 



