496 Prof. Donkin on the Geometrical Interpretation 



script, and use + simply, to denote a x<chole revolution in any 

 plane. But there are no other values of a besides these two 



of a=l, a.= ~y and their multiples, for which the effect of the 



rotation ( +,.)" is independent of the direction of r. It is also 

 to be noticed, that these ai*e the only cases in which the 

 symbol ( + ^)" \s commutative with respect to all other symbols 

 of rotation. The reader will easily see, that, on the principles 

 we have admitted, ( + ^)*2' is equivalent to 2^( +,.)*. Either will 

 be henceforth denoted by —q. 



Now let there be three fixed rectangular axes meeting the 

 sphere in X, Y, Z, and so arranged that X is the north pole 

 of the rotation YZ. Let us for convenience assume three 

 fixed symbols, /,j, ^, to represent the three rotations YZ, 

 ZX, XY; that is, quadrantal rotations respectively in (ciny 

 parts of) the three planes of lyz, zx^ xy. According to our 

 previous notation, we might write 



^■=(+.)^ i=(+,)^ ^'=(+J*; 



but we shall not have much further occasion for it. 



The conclusions obtained above respecting quadrantal tri- 

 angles give the equations 



jk=i, Id-j, ij=k, Jcj- -J, ik=-., ji--T\ 



I J K 



of which the three last can now be written 



kj= — i, ik=-j, ji——k. 

 For we have 



and we have seen that —„ may be replaced by — , so that we 

 get 2T'= — z, and, in like manner, 



Also we have 



and similarly. 





which may now be written 



and, to conform to common notation, we may put —1 instead 

 of — . In all these cases it is to be recollected that the symbol^. 

 = denotes, not identity of operations, but equivalence of 

 results. odfiadJ'io 



