444 Mr. H. T. Flint on the Applications of 



If Kitf denotes the quaternion conjugate to iv and P is the 

 force quaternion, the equivalent of (xiii.) is 



SP.Kto = (xiv.) 



This is the same as the condition for constancy of internal 

 energy given in the ' Theory of Relativity ' *. 



The quaternion Kw or any physical quaternion of the form 



is transformed to S / by the operation Qc^Qc, where Q c is 

 derived from Q by writing — v instead of v f. 

 Transformed to S', SP . Kw becomes 



SP'K™' = SQPQ . QcKioQ c =SQPKk;Q c . 



Thus PKio is an " P " quaternion J whence its scalar is 

 invariant. Thus SP'Kw'=:0, or the principle of energy is 

 invariant. 



7. From (xiii.) we obtain a more general result by 



regarding the term on the right, viz. -=- -t~(~t ) prefixed 



with the negative sign and multiplied by m as the rate of 

 change of energy, i. e. 



~r (ener 



oil d [ dl\ 



This leads to the expression ^m k 2 for the energy, omitting 

 an arbitrary constant. 



We may denote the kinetic energy by the expression 



It has been pointed out by Jeffreys || that while there is a 

 certain arbitrariness in the choice of the exact form for the 

 kinetic energy there is convenience in the adoption of this 

 form. 



This expression, like m (k— 1), reduces io the ordinary 

 value \mv 2 for velocities very much smaller than that of 

 light. 



* Cunningham, Theory of Eel. p. 167. 



t SilbersteiD, Phil. Mag. May 1912. 



% Silberstein, ibid. 



§ Cf. W. Wilson, Proc. Phys. Soc. xxxi. pi. ii. p. 74. 



r | H. Jeffreys, Phil. Mag. July 1919. 



