2 QUATERNIONS APPLIED TO PHYSICS 



theorems are true without any real modification (so that 

 for instance a vortex theory of matter is just as applic- 

 able in non-Euclidean as in Euclidean space) ; (2) that 

 except when wav^e-lengths are infinitesimal compared 

 with the space constant, only one of the four types of 

 wave-motion mentioned possesses velocity independent 

 of wave-length (and therefore possesses equal velocities 

 of propagation for waves and for groups of waves). 

 The one of the four is wave-motion in ether. 



When below we come to byperbolic space we shall 

 use complex quaternions [p + p' s/( — 1) where p and/;' 

 are real quaternions]. If ^ is a complex quaternion, 

 then, as Hamilton prescribed, the tensor Tq and the 

 unitat U<7 will mean T^- = sj {qYiq) and XJq = qTq, 

 with the proviso that the scalar Tq is that particular one 

 of the two square roots whose real part is positive ; 

 or if the real part is zero, Tq is I multiplied by a posi- 

 tive scalar, I standing for >/(—!). The arc Aq of q 

 will mean a complex scalar o + la = cos~i SU^ where 

 a and a are real, a ranging between and tt, and a 

 between -f- go and — oo ; and when a is zero a is positive. 

 a will be called the angle of q and a the advance. When 

 the advance is zero the unitat is a versor ; when the 

 angle is zero the unitat is a translator ; when neither 

 angle nor advance is zero the unitat is the product 

 of a versor and a co-axial translator. 



Elliptic Space Preliminaries, 



§2. The quaternions to be applied in elliptic space are 

 never complex ; real quaternions furnish a complete 

 geometric method, and Clifl^ord'sbi-quaternions naturally 

 come forward with a second allied method. For the 

 present the quaternions are to be real ordinary quater- 

 nions. 



Let, in the first instance, the axes of a complete 

 system of quaternions be lines through a fixed point O 

 in an elliptic space, and let any quaternion q of the 

 system (in addition to its usual signification of a quotient 

 of two vectors through O) signify a length measured 

 along the axis of Nq equal to A^'. If C)P is in this 

 direction and of this length, q may be called the position 

 quaternion of P (origin O) and U^ may be called the 

 position unitat of P. Next assign yet another meaning 



