Quaternions to the Theory of Relativity. 441 



Write q = & + \ 



so that a' + V = Q(a + \)Q. 



It follows that 



a' = \(l-/3^v+a + v(l-/3)Sav . . . (i.) 



and \' = £\ + (l-£ 2 )*Sav (ii.) 



2. From the quaternion q we pass to 



d 



We may describe w as the velocity quaternion. It is 

 evidently a physical quaternion, for dr is invariant. 

 This statement, or what is the same thing, 



contains all that need be said about transformation of 

 velocities. It is, however, interesting to derive the well- 

 known formulae. We have 



Thus by (i.) and (ii.) 



^ = J(l-^v + ^r + v(l-^)S^ r .v,. (iii.) 

 and 



fw^+a-w** (iv.) 



From (iv.) 



L_ g vj3 dr f . 



(l-V 2 )* ~ (1-u 2 )* + {i-v?)^TTt mY > ' (v,) 



and if v is taken along O.r this gives 



(tEJ)*^ 1 - 1 '"*) ( vL ) 



This is the well-known and important transformation of 

 (l-u /2 ). 

 From (iii.) by making use of (vi.) 



! Ux — V 



1 — VU X 



