100 Proceedings of the Royal Irish Academy. 



vectors in their plane (/i andja), provided the inclination of the new 

 vectors is equal to that of the old. Similarly, a^ and aj may be replaced 

 hy vectors coplanar with them (a'2 and a'l). The operator is now 



j^oa'^a\ ( ) a'r^aWa'V'rS 



and if/o and a'2 are taken to be along the common line of intersection 

 of the planes, the operator reduces further to j/'ia'i ( ) a'f ^f^ 



Thus, it is proved that it is possible to determine an operator 

 A^i ( ) ySr^A~S which will produce the same effect on vectors co- 

 spatial with tti, aj, and a^ as the operator a3a2ai ( ) ar^a2~^a3~^, though 

 the first preserves, while the second reverses, the directions of vectors 

 perpendicular to the space. 



35. It is instructive to contrast and compare the two operators 



(B,IB, ( ) ySr^A-S and nuhlio(S, ( ) ^r/S.-h-fSfH,-^ 

 in greater detail. 



As «i, 4, and 4 may be any triad of units in the given space, 

 suppose 



^1 = ?'i, and ^2 = h cos u + f 2 sin u, 



and then /Sz^^ = - gosu - iitz sin u, 



while iiHhP^^i = - hhh cos u + is sin u. 



The essential elements of the two operators are presented in two 

 different ways. The first involves the angle u, and the symbol (iiii) 

 of the plane in which (or parallel to which) the rotation through the 

 angle 2u takes place. The second involves u, the symbol of the space 

 («i4«3) containing the plane of the I'otation, and that particular perpen- 

 dicular (?'o) to this plane which is unaffected by the operator. Of course, 

 from ^3, and the product h/s^) the symbol of the plane (h/2) inay he 

 deduced. 



36. It is desirable to show that, by an operator of the type q { ) q~^, 

 any set of vector units, ii, i^, . . . im, iiiay be converted into any other 

 set, Ji,j2, • • -Jm- The new set is not necessarily in the same space 

 of m dimensions. Indeed, generally a space of 2m dimensions will be 

 required to contain both sets. This investigation will also be useful 

 for other reasons. 



The vectors being unit, obviously 



ii +j\ = (1 -j'iii) ii =ji (1 -jiii), 



and so the operator (1 - j\i]) ( ) (1 -jii\)~^ will convert ?i into/i, but 

 will leave unchanged any vector perpendicular to both. For brevity, 



