6 Proceedings of the Roijal Irish Academy. 



The quaternion is, consequently, reduced to 



q (<^i, ^2, ^3) = 2 (^11723 - <^2'73i + '^z'nn) + k {<ku <^2, ^3), 



and is therefore, as has been announced, independent of the vectors 

 a, P, and y. 



Occasionally, the vector 2972s may be designated by 



but care must be taken to distinguish between 



<i>l ^{4>3 4>2 - <}>i^3) • P = 2<^l')723 . P, 



and 



^1 (</>3'^2 - ^2'^3) P = 2^1 Vrjizp. 



7. Special cases of this invariant. — As a particular case of the 

 preceding invariant, let ^1 = ^3, and then 



For 



^21 + >7i2 = 0, and yjn = 0, 



and the vector part vanishes. This might have been predicted, from 

 an example in Art. 5. If ^3 = t^g, 



S (^15 <^2> ^2) = 4 {<i>ii ^2, ^2) + 4<^3'>7i2 ; 

 hence, in particular, if <^2 = I5 



^(<^i, 1, l) = h{<l>i, 1, l) + 4ei; 

 in which expression, remembering that 



ci is the spin -vector of <^i. 

 Now, 



I3 (^1, 1,1) = 2^1 if ^1' - Wi<^i^ + mzcjii - JWj = 0, 



and, therefore, 



^(<^i, 1, l) = 2(wi + 2e0, 



which is Hamilton's first invariant. Also, in a similar manner, 



9 (<^i> ^u 1) = h (<^i, «^i, 1) + 4<^i€i 

 = 2 (mj + 2(^iei) ; 



and this is Hamilton's second quaternion invariant.^ 

 S {^D ^h ^1) is easily seen to be equal to dm^. 



1 See Ma "Elements of Quaternions," Art. 349. 



