

[ 389 ] 



XXXIV. Integration Theorems of Four-Dimensional Vector 

 Analysis. By H. T. Flint, Lecturer in Physics, King's 

 College, London *.. 



Introduction. 



THE four-dimensional vector analysis developed by 

 Sommerfeld {Ann. der Physik, vols, xxxii. p. 749 sqq* 

 and xxxiii. p. 649 sqq.) evidently bears the same relation 

 to a more general analysis as do scalar and vector products 

 and the theorems of Gauss and Stokes to vector or quaternion 

 analysis. 



In attempting to work out some of the details of the 

 restricted principle of Relativity in terms of the more 

 general notation, it was observed that such expressions as 

 the scalar and vector products of four vectors, combinations 

 of four and six vectors, together with the generalized div 

 and curl and the operation lor (Lorentz operator) of 

 Minkowski's Calculus are parts of more general expressions. 



It is the object of the following pages to set forth this 

 analysis. It will be obvious to the reader how much is owed 

 to quaternion analysis, in particular to Joly's ' Manual of 

 Quaternions/ ch. xviii., of which the notation is adopted here, 

 and to Sommerfeld's papers mentioned above. The applica- 

 tion of the notation to the theory of Relativity has already 

 been pointed out by Professor Johnston and discussed by 

 Sir J. Larmor t- 



In the first few sections the notation is explained before 

 the main object of developing the integral theorem is 

 reached. 



§ 1. In treating space of four dimensions it is usual and 

 convenient to adopt a language and notation similar to that 

 with which we are familiar in three, though it is difficult 

 and perhaps impossible to form corresponding geometrical 

 pictures. 



Thus a straight line through the points (a 1 b l t\ d{) 

 (a 2 b 2 c 9i d 2 ) is the locus : 



x — a x __ y-b 1 __ z-c-x _ u-d l ^^ 



a x — a 2 b l — b 2 c 1 — c 2 di — d 2 ' 



(x y z u) being current coordinates, axes O.i', 0//. 0.\ Om 

 being assumed as axes of reference. 



* Communicated by the Author, 

 t Proc. Roy. Soc. Dec. 1919. 



Phil. Mag. S. G. Vol. 41. No. 243. March 1921. 2 1> 



