FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
401 
Quaternionic Form of Stress- Vector. 
§25. We may also notice the Quaternion form for the stress function, which is so 
vital a part of the mathematics of forces varying as the inverse square of the distance, 
and of potential theory. Isotropy being understood, the electric stress may be 
written 
P N = ic[EN- 1 E],.(172) 
where the quantity in the square brackets is to be understood quaternionically. It 
is, however, a pure vector. Or, 
fPV 
c 
-E- 
_E _ 
= 2 
_N_ 
(173) 
rp . 
that is, not counting the factor 1, c, the quaternion is the same as the quaternion 
; or the same operation which turns N to E also turns E to P N . Thus, N, E, 
and P N are in the same plane, and the angle between N and E equals that between 
E and P N ; and E and P N are on the same side of N when E makes an acute angle 
with N. Also, the tensor of P N is U, so that its normal and tangential components 
are U cos 2 6 and U sin 26, if 6 = HE. 
Otherwise 
P N = -£ C [ENE], 
(174) 
since the quaternionic reciprocal of a vector has the reverse direction. The corre¬ 
sponding volume translational force is 
F = — cV [EYE],.(175) 
which is also to be understood quaternionically, and expanded, and separated into 
parts to become physically significant. 1 only use the square brackets in this 
paragraph to emphasise the difference in notation. It rarely occurs that any 
advantage is gained by the use of the quaternion, in saying which, I merely repeat 
what Professor Willard Gibbs has been lately telling us; and I further believe the 
disadvantages usually far outweigh the advantages. Nevertheless, apart from practical 
application, and looking at it from the purely quaternionic point of view, I ought to 
also add that the invention of quaternions must be regarded as a most remarkable 
feat of human ingenuity. Vector analysis, without quaternions, could have been 
found by any mathematician by carefully examining the mechanics of the Cartesian 
mathematics ; but to find out quaternions required a genius. 
