17 
1921-22.] On Tubes of Electromagnetic Force. 
Now take {hx, hy, hz, dx, dy, dz) to be the components of the magnetic 
and electric force at a point in free space : then by Maxwell’s equations (2) 
the triple integral on the right-hand side of de Franchis’ equation vanishes ; 
that is to say, if, in our four-dimensional hyperspace, S he any closed 
surface to which we can fit an open hypersurface not containing elec- 
trons, then the integral 
III 
d{x, y) d{x, z) , ^ d(x, t) _ d(y, z) 
V) ^d{u, v) ^d{u, v) v) 
. t) 9(2;, t) I . ^ 
( 18 ) 
vanishes when the integration is extended over the surface S.^ 
Similarly, by use of the other four of Maxwell’s equations (namely, 
equations (1)) we can show that the integral 
^ 9(x, y) 9(a;^) d{x, t) d(y, z) d(y, t) , d{z, t) 
^d{u, v) ^d{u, v) ®0(w, v) ^d{u, v) v) v) 
I du dv (19) 
vanishes when the integration is extended over the surface S. 
These two formulce (18) and (19) constitute an integrated equivalent 
of Maxwell’s equations in free space. 
§ 12. The Integral Properties of Tubes of Force. 
We shall now apply the formulae of § 11 to the case when the closed 
surface S is formed of a portion of a thin tube of force or calamoid, 
terminated at one end by a portion cr of an electropotential surface, and 
terminated at the other end by a portion r of another electropotential 
surface. 
By the equations (16) the integrals vanish over that portion of the 
surface S which is formed of the calamoid : and therefore we have the 
result that the integrals 
III 
d(x, y) d{x, z) d{x, t) 
^d{u, v) ^^0(w, v) *0(w, v) 
f , j t) _ ^ d(z, t) ( 
+ d. 
and 
III 
j d(x, y) , ^ 0(;r, z) , d{x, t) 
%{u, v)'^ vy ^d{u, v) 
d{u, v) '^d{u, v) v) 
,1 KyV) , 8(z, t) 
^d{u, v) '^d{u, v) ^d{u, v) 
du dv 
du dv, 
vanish when the integration is extended over the two small regions 
<j and T together. 
Now, since cr is part of an equipotential surface, we have by equation (7) 
dfipjf) d(x, z) d(x, t) d{y, z) d(y, t) d(z, t) 
d(u, v) _ 0(?i, v) _ d{u,v) _ d{u, v) _ 0(?/., v) d(u, v) 
dg d,y h X dx hy 
* This result is not new, and is inserted here, only because it is necessary for what 
follows in § 12. 
VOL. XLII. 
2 
