19 
1921-22.] On Tubes of Electromagnetic Force. 
First, we may observe that the formation of surfaces by the aggrega- 
tion of curves is a very common phenomenon in connection with the 
solution of partial differential equations. Thus, in order to solve the 
linear partial differential equation 
we first solve the sj^stem of ordinary differential equations 
dx _ dy _ dz 
P "" 13 "" K ' 
The integrals of this latter system represent 00 ^ curves, one curve 
passing through each point of space : these are called characteristic 
curves. Any surface which is built up of 00 ^ of these characteristic 
curves, associated according to any law, is an integral -surface of the 
partial differential equation : and any two integral-surfaces of the partial 
differential equation intersect in one of the characteristic curves. 
Characteristic curves exist also in connection with partial differential 
equations of the first order which are not linear : but for these non-linear 
equations the integral-surfaces which pass through a characteristic curve 
must touch each other all along it, so that with each point of a character- 
istic curve is associated a surface-element : the curve with its associated 
surface-elements forms a ribbon-like strip, and it is of these strips that 
the integral-surfaces are formed. 
In order to obtain a theory similar to this for the calamoids, we write 
dz dz dt dt 
^=92- 
and write 
Pi(h~V2di = ^' (20) 
so that the partial differential equations of the calamoids are, by (17), 
hz hyti^ -f d^c^ tixPi dyP2 + — 0 ) 
— d.^ + dyq^ + P dxP\ ~ + tip' = 0 J 
( 21 ) 
The tangent-plane at {x, y t) to a calamoid has the equations 
I -py^-x)-q^{Y-y)PiZ-z) = 0 
I - x)-qj^Y - y) + {T -t) = Q , 
so for any line-element {Sx, Sy, Sz, St) in this tangent-plane we have 
-p^Sx-q^Sy-pSz = 0 i 
-p^dx - q^Sy -pSt — 0 ) 
Let us consider the case when the calamoid intersects a neighbouring 
