JoLY — TJie Associative Algebra applicable to Ri/perspace. 9f) 



is the result of tiiis double reflection. It is manifest, geometrically, 

 that the component p' is turned by this operation through twice the 

 angle between a^ and a^ in the plane of these two vectors, and in the 

 direction from aj to aj. 



It is also evident that the essential elements in this operator are — 

 (1) the plane of ai and a^, (2) the angle between a^ and as, and 

 (3) the direction of rotation from aj to a^. It is clear that the lengths 

 of the vectors ai, aj, and their absolute positions in the plane are not 

 essential, and therefore that the operators 



Oatti ( ) ai"'a2"S aild a'sot'i ( ) a'r'aV^ 



are equivalent, provided the accented vectors are coplanar with those 

 not accented, and the angle between ai and a^ is equal to that between 

 a'l and a 2, when these angles are measured in the same direction. 



It will be noted that the operator here considered is without effect 

 on any vector perpendicular to the plane of a^ and 02. For, if /? is any 

 such vector, 



azO-ifi = — a2;Sai = (3a2ai. 



34. IText, consider the operator ajasai ( ) ar'aa'^aa"^ which reflects 

 a line successively to aj, ao, and a^, but which reverses the direction of 

 every vector perpendicular to these three vectors. Supposing that the 

 vectors a are not coplanar, let ?i, ^2, and ^ be any three mutually 

 rectangular nnits in the tri- dimensional space determined by them. 



It is evident, by the law of interchanges, that the operator 



reverses the direction of every vector perpendicular to that space, and 

 produces no change on any vector contained in it. 

 Hence, it appears that the operators 



aattjai ( ) af^ ch^a.^^, and «i44a3a2ai ( ) oLi~^a-2~^a-z'^ii^i2'^i{''^ 



have the same effect on any vector contained in the above-mentioned 

 space ; but the first reverses, while the second leaves unchanged, the 

 direction of any vector perpendicular to that space. 



Now, without loss of generality, ^ may be taken parallel to 03 ; in 

 this case, the second operator reduces to 



e'l^ajai ( ) af ^ao^^v'^'r^'- 



Since all the vectors considered lie in the same space of three dimen- 

 sions, the planes of ?i, i^, and of aj, ao will intersect in some common 

 line. By the last article, ?'i and iz may be replaced by any pair of 



H2 



