Quaternions to the Theory of Relativity. 443 



Writing F x = ~r(m u x ) etc. we find from (viii.) 

 at 



-^ dm 1 



h x —v — 



F,'« ^ — =F ar - 1 -^- F y - ^-^^F, 



1— tni x 1 — vu x y 1— vm x )>.(xi.) 



and similarly, 



F '= -^ F ' = 



y /3(l-iO' * ${l-mi x ) j 



These are Planck's equations for transformation of force. 



6. If mass be regarded as a manifestation of contained 

 energy we may, on this view, regard m as a measure of 

 the energy of a body at rest. 



The expression for the energy is — — ^\i • Thus the 

 dl . * [± — u)z 



scalar term m — of M is equal to i (energy) . 



From the definition of dr we have 



(dT) 2 =(dvy 2 -(di) 2 (xii.) 



Hence /dv\ 2 (dl 



an 



or 



d 



dr\dr) ~~ dr\dr) ' 



dx d 2 v dl dH , ... , 



Multiply throughout by m and the term on the left becomes 

 1 d_ f dr) 



(i-^ 2 ) dt\ m dtj' 



On the right we have 



1 d f m ) 

 (1-m 2 ) ^ ( (l-ti»)*J ' 



Thus 



*dt 'dt'\ m 7t)~ dt' X (l-u 2 )i J 



This equation represents the principle of conservation of 

 energy, for on the left we have the activity of the force and 

 on the right the rate of change of energy. 



