104 Proceedings of the Royal Irish Academi/. 



Operating on this by (f>, the result is 



X (<j''(«+«')^ + ^-"("t")^') = a cos (m + v) + a' sin {u -\- v). 



So the operator <^ merely turns any vector in this plane through an 

 angle equal to u. 



40. In the last article it was shown that a pair of conjugate roots 

 and axes of the function ^ may be expressed by the equations 



^ (a + ha') = e''" (a + Ita'), and ^ (a - ha') = e"''" (a - ha'), 



a and a' being jDcrpendicular and real vectors of equal length, and h 

 being the imaginary of algebra. 



For a second conjugate pair distinguished from this pair by the 

 suffix 1, the relations 



show that, if w is not equal to + u, it is necessary to have 



S/3^, = >S/3'y8i - 8/3/3', = S/3'/3', = 0. 

 Hence, it is necessary to have 



Saai — Saa'i = Sa'a, = Sa'a'i = ; 



or both the vectors a and a' must be perpendicular to a, and a'l ; or the 

 planes of a and a', and of ai and a'l are hyper-perpendicular. 



Hence, it is possible to obtain a clear perception of the properties 

 of the operator q{) ^^ which converts vectors into vectors. In con- 

 nexion with any such operator there exists a certain number of 

 hyper-perpendicular planes, and the operator turns the components 

 of a vector in each of these planes through certain definite angles, 

 different in general for each plane. 



If the operator involves units contained in an w-space, there are 

 ^m or -g- (ot - 1) such planes, according as m is even or odd. For an 

 odd space, there is one common perpendicular to all these planes, and 

 the operator has no effect on this line. 



41. Prom the last article it appears that the general rotation of a 

 rigid body of m dimensions in w-dimensional space may be resolved 

 into rotations of definite amounts in ^m or ^{m- 1) hyper-perpen- 

 dicular planes. 



ISTow, a rotation in the plane of iii^ may be represented by 



2'i2 ( ) qi2~\ where $-12 = cos 1^12 + iiii sin ^u^^ ; 

 and it is easy to see that 



^12"^ = cos ^Mi2 - ^'i«2 sin ^Ui2. 



