81 
1919-20.] The Quaternionic System and Physics. 
As an example, we may consider Johnston’s problem, 
sional operator 
V 
. 0 . 0,0 
The four-dimen- 
has to be modified to suit a tridimensional intelligence. As above, the 
substitution for l is O ijk. The quantity u is a four^dimensional scalar 
whose numerical magnitude is ct. If u be an imaginary scalar, the 
operator remains homogeneous if l be made imaginary. It may then be 
replaced by O. Thus 
V fi = v- +j~ + k~ + c 1 n~ 
dx dy dz dt 
Operating by this upon Johnston’s vector U written in the form 
c(&F +yG- + A:H) — Q<f) , 
where the vector in the bracket is the vector potential and </> is the scalar 
potential, we find 
QV7 TT /03f 0Gr 
S V Q IJ= -c(— + — 
0H\ 
dz ) 
- c 
dt 
Y . V nU = jk . a + /ci . (3 + ij .y + (idX. +JOY + kU,Z) 
(a, f3, y) and (X, Y, Z) being the magnetic and electric force components 
respectively, each measured in their own distinctive units. 
The vanishing of S . VoU gives the electrostatic condition for the dis- 
tribution of electricity and electric force. The vector part is a Minkowskian 
“ six-vector.” Operating upon it by V 0 we get 
VeVVoU 
0a 0/3 0y 
dx dy + dz 
0X 0Y 
dx dy 
;/0y_0£ 
\dy dz 
+ . . . 
the vanishing of which gives the conditions obtaining in the electro- 
magnetic field in free space, and exhibits them as a consequence of the 
condition of relativity, as Johnston has shown. 
In analogy with the previous procedure, if we had to deal in three- 
dimensional space with some effect in five-dimensional space in which the 
more limited space existed, we should have to introduce 0 5 , where 
VOL. XL. 
ijklm - - - Q 5 . 
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