18 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
and each of these fractions is equal to 
y) V g)P / %> ^) V i 0 \ 
_l0(z/, v)) v)j \0(«^, v)J l0(w, r)i \0(w, v)\ l0(i!i, v)} _ 
But the area of <j is 
~f 0(x, y) \^ f d(x, z) \^ (d(y, z) f d(x, t) \^ t) ^ du dv 
_\ 0 ( 2 ^, v)} l0(?4, v)J \0(t^, v)^ \0(w, v)) \0('^/, v)i \0(w, r)J J ’ 
the symbol of integration being omitted, as o- is a very small area. 
Thus we have 
y) 
d{u, v) 
du dv 
X area of 
and therefore the first integral-formula becomes nugatory, while th 
second integral-formula transforms into the theorem that the area of a 
multiplied by the value of {dx^-Vdy^^-^-dz^ — hx^ — hy^ — hz^fd^t cr, is equal to 
the area of r, multiplied by the value of {dx^-\-d^-\-d^ — }ix-‘ — hy-‘ — li^-)^ at t 
I n other words, the cross-section of a thin calamoid (measured by the area 
which it cuts off on the electropotential surfaces which intersect it in 
curves), multiplied hy the value of (dx^ -f dy- -h df — hx^ — hy^ — is con- 
stant along the whole length of the calamoid. 
It will be seen at once that this theorem is the generalisation of the 
well-known property of Faraday tubes of force in electrostatics, namely, 
that “the cross-section of a Faraday tube, multiplied by the value of the 
electric force, is constant along the whole length of the tube.” The 
quantity {dx^ + dy^-\-dz^ — hx^ — hy‘^ — hz^f, which occurs in the corresponding 
property of calamoids, is independent of the velocity of the observer by 
whom the electric and magnetic forces are measured, and is therefore 
suited for the expression of an invariant property. 
§ 13. The Characteristics. 
The Faraday tubes of force in electrostatics and magnetism are 
constructed in the following way. We first take the system of curves 
which intersect the equipotential surfaces orthogonally ; these are called 
the lines of force : then we take any simple closed curve in the field, 
and consider the lines of force which intersect this curve : they form a 
tubular surface, which is called a tube of force. A tube of force, which 
is a surface, is therefore formed by the aggregation of lines of force, 
which are curves. 
We have now to inquire if there is anything analogous to this in the 
theory of calamoids. 
