106 Proceedings of the Royal Irish Academy. 



<fi - 1 generally reduces to zero some vector a ;^ so, if 8 has a coia- 

 ponent parallel to a it cannot be removed. 



Thus, in spaces of even order, the general displacement of a body 

 may be effected by rotations of definite amounts in a number of definite 

 hyper-perpendicular planes, one determinate point being held fixed ; 

 in spaces of odd order, a translational displacement must be added 

 to the generalized rotation ; but by proper choice of base-point this 

 displacement may be made perpendicular to all the planes of rotation. 



This is completely analogous to the displacement of a body in two 

 and in three dimensions. 



43. A new form may be given to the operator Q{ ) Q'^, which 

 clearly exhibits its essential elements. 



The expression §■12 = cos ^ti^ + hiz sin -0-^12 t^^J be written in the 

 forms „j2 



qn = e5'i'2"i4, or qi^ = iJiiiY^- 



Thus, Q = e^(v>i. + v.»3*+- • ■\ 



or Q = (/1/2) T {iiii) TT . . . &c., 



for the products /i^j 4u, &c., are all commutative,^ so that it is allow- 

 able to write 



Again, if §'2 is any quadratic function of the units, e^2 ( ) e'^^ is an 

 operator which will produce a conical rotation of the general kind ; 

 this is because it has been shown in Art. 10 that the general quadratic 

 function may be reduced to the form 



q2 = (^liiiiz + (^3ihii + . . •, 



and in (?'= ( ) e"^^, the coefficients ai2, (isi, &c., are double the angles of 

 the rotations in the corresponding planes. 



In the limit, if the rotation is infinitesimal so that thh angles, 

 and therefore §'2, are very small, 



e^2pe-i2 = (1 + 22) p (1 - 32) = p -)- {q-p - pq^) = p + 2 Fi22p, 



and this agrees with the expression used in Art. 8. 



The essential elements in these various expressions are, of course, 

 the angles and the symbols (/1/2, v'l, &c.) of the fundamental planes. 



1 See Arts. 38 and 40. 



- See Elements of Quaternions, Ail. 316 (20). 



