2 Proceedings of the Royal Irish Academy . 



the Eelativit^" piinciple as developed by Lorentz, Lamior, and especially by 

 EinsteiiL* 



It is the pnrpose of this Paper to show that a quatermon method can be 



employed which gives the required conciseness to the resnlts, and which 

 readily indicates new theorems. The notation of Hamilton is employed 

 throughout, but for the purpose of clearness clarendon type is employed for 

 qnatemions. The Operator Af corresponds to v, but is supposed to act on 

 every part of the term in which it occui-s. The algebraic imaginary \/ -1 

 is denoted by /i.x 



§ 2. FCSDAilESTAL FOBMTT T.AK AM) ESAiCPLES. 



If £ denotes the electric force, and j| the magnetic force, we have the 

 fundamental equations in free aether, 



c'i = V), 

 - ^ = Vf- 



the imits being electrostatic. It follows that the constant c is the speed of 

 radiation. If we multiply the upper equation by lie, and add it to the lower, 

 both equations are included in the single equation 



(V - hc-^dldt) ii + hen) = 0. 



If we denote the biquatemion (in Hamilton's sense) operator V - lic^djdt 

 by (J, and the biveetor t + hcif by <r, we have the equations expressed in 

 the form§ 



(Z-<r = 0. 



If there is pi-esent a current i and a volume-density c, the fundamental 

 equations are 



-v=rv,, 



together with the equation of continuity 



e = ,ST.. 



* Einstein : Annalem der Phrsifc, 17 and 18 (190-5), 20 and 21 (1906), 23 (1907). 



t M'Aulay : The TTdliry of Quaternions in Physics, p. 13. 



{ Hamilton's Elements of Quaternions, toL L, p. 290. 



§ The biveetor is employed by Weber : Die partiellen difierentialgleichungen der math. Physik ; 

 and by SiLberstein : Annalen der Physik, 24 (1907). The further step of using the biquatemion 

 was given by the author at the British Association Meeting at Dublin (1908). An equivalent 

 formula, but expressed in the matrix notation, was given by the late H. Minkowski in a remarkable 

 paper in the Ifach. Gottingen (190S), p. -53. In E. B. Wilson's edition of Gibbs' Vector Analysis 

 it is stated that Gibbs used the biveetor in his lectures on Physical Optics. Many of the properties 

 of the operator (j were given by the late Prof. C. J. Joly as questions in a Fellowship Examination 

 in the Riyal University (1900). 



