14 QUATERNIONS APPLIED TO PHYSICS 



present themselves, as in most physical applications, then 

 equations after the model of (1), which treat the motors 

 as wholes, are to be preferred to the others. But in 

 pure geometry, where three scalar coordinates, in place 

 of six, are more natural, the method of §2, and the 

 corresponding method for hyperbolic space, namely 

 Joly's, seem to the writer superior. 



If is a quaternion linity, the conjugate ^' of (p is 

 defined by S^^r = Sr^'^y for any two quaternions q, r. 

 The K-conjugate ^' of is defined by S.qK(pr 

 = S.rK(p^q ; that is, the bilinear scalar S.pK.q is used 

 in the definition of ^' in the same manner as the bilinear 

 scalar ii.pq is used in the definition of ({/. It follows 

 from the definitions that 0' = K(^'K ; K itself b«ng a 

 quaternion linity which is both self-conjugate and self- 

 K-conjugate. In elliptic space 0^ is of importance, and 

 not (j)'. Tins is due to the fact that the equation of the 

 absolute is S.qKq = 0, not S.y^ = 0. The equation of 

 any quadric is S.qKipq = where (p is self- K-conjugate. 

 The quadric being real, and therefore <p real, it can, by 

 proper choice of origin invariably be expressed in the 

 (wholly real) form 



0^ = ^ Sy — a/'S/'q — ^ijSjq — c/iSkq 

 i, /, k intersectmg in the origin. [For an application 

 below it should be added tliat when we do not permit 

 choice of origin the real form is 



1>^J^^(f/^-'J]^~^ ~ ftfS.iqp'^ — f'jSjqj>~^ — ck^.kqj)'^). ]>, 

 Putting p = ?'p' the last changes to 



(Pq = {aS.qp-^ — r/fS./qp-^ — ('j^'jqP~^ — bh^.kqp-^).p\ 

 there being no real distinction between the four mutually 

 quadrantally distant points whose position unitats are 

 1, i.j, h]. 



The general real K-skew linity (^' = —(f) has the 

 very simple form (pq = aq + 7/8 where a, j3 are real 

 given vectors. The general (p for Avhich ^'^ = 1 = ^(^^ 

 has the equally simple form 



^q = pqp'''^, with T/^ = Ty/. 

 In elliptic space the point (or plane) transformation 

 r ^= fpq, (p''(p = I, means kinematically the general finite 

 twist; and the transformatijan r= (pq, ^^ = — ^, con- 

 Tcrts a position quaternion q into the rate of change of 



