WILSON AND LEWIS. — RELATIVITY. 413 



If we had chosen a diU'orent set of perpendicuhir axes k/, k4', where 

 ki' makes an angle \{/ = tanh"^ u with k4, so that the inclination of w 

 to k/ is <^' = — 1^, the new components of w would be 



"7^ = sinh <^' = cosh 4) cosh i/' — sinh (j> sinh \p = — = 



ds Vl — v'^ 



V — M 



Vl— u2 Vl— m2 

 J / 1 



—^ = cosh \f/' = cosh (/) cosh 4' — s^i^l^ <A sinh \p = , - 

 a* Vl — 1)^ 



1 — vu 



Vl _ «2 Vl — u2 

 where 



dxi 1 — tanh </> tanh i/' 1 — vu 



(6) 



It will be convenient to have a general equation for the components 

 of a vector upon one set of axes in terms of its components on another 

 set. Let ki, k* be one set of perpendicular unit vectors, and k/, 

 k/ another set. If the angle from ki to k/ be \{/, the angle from k4 

 to k/ is also i/' by § 16. The products 



ki'ki' = cosh^i', k4'k4'= — cosh;/', 



ki'ki' = sinh^, ki'«k4 = — sinh;/', 



follow from (1). To obtain the transformation equations we write 

 r = .Tiki + Xiki = .r/k,' + •^4'k4', 



and multiply by ki, kj, k/, k4'; 



r-ki = .Ti = .t'l'coshi/' + .1*4' sinh ;/', 



— r«k4 = .T4 = .ri'sinhi/' + .T4' cosh 1/', 

 r-ki' = xi = .ricoshi/' — .1-4 sinh i/', 



— f •k4' = .Ti' = — .t'l sinhi/' + .T4Cosh^. 



Curvature in our non-Euclidean geometry' is defined, as is ordinary 

 geometry, as the rate of turning of the tangent relative to the arc. 

 As W is a unit tangent, dw is perpendicular to W and in magnitude is 

 equal to the differential angle through which w turns. Hence 



