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



moving with velocity, v, with respect to a system, S, along 

 a direction denoted by the unit vector v. 



The vector part of q' is r', the vector from the origin to a 

 point P', and the scalar is V where V — it\ t' is the time in S' 

 and is measured in units in which the velocity of light is 

 unity, q is similarly r + Z, in S. 



Q is also a quaternion, and expressed in detail is 



-^ {(r+£)i + (l-/3)iv} 



where /3 = (1 — v 2 )~*. 



It is to be applied in front of and behind the quantity to 

 be transformed and obeys, of course, the rules of multiplica- 

 tion of quaternion analysis. 



Expressions like </, which transform in this way, are called 

 physical quaternions, and evidently such quantities, like 

 four - vectors, are capable of expressing the theory of 

 Relativity, in fact, they are just what is required. 



We here consider the application of this notation to 

 velocity, force, and momentum. 



It will be seen that the well-known results of the 

 Cartesian mode of expression are easily derived, but the 

 results obtained are more general and have no special 

 reference to axes. 



An expression for the kinetic energy of a particle, slightly 

 different from that usually accepted, is indicated by the 

 notation. This form has been discussed elsewhere by 

 W. Wilson *. 



Application to electric and magnetic forces give quite 

 general transformations, and we again recognize by refer- 

 ence to special directions the Cartesian formulae resulting. 

 Finally, the problem of the field due to a uniformly moving 

 charge is solved by a very easy application of the general 

 formulae. 



1. Notation. 



t will be used in its usual meaning, so that 



(dr) 2 = - {dx 2 + dif + dz 2 + dl 2 ) = (l-u 2 )dt 2 = ~dt 2 (say) 



■ -m-m-m 



dr is an element of the " proper time " and is invariant in 

 our transformations. 



* Proc. Phys. Soc. xxxi. pt. ii. p. 74. 



