21 
1921-22.] On Tubes of Electromagnetic Force. 
Substituting from (26) and (27) in (25), we have 
(N - M){(Aa, + + {hy + qi^h)^y}{(<fx + Pid^)8x + (dy + qi^d^)S7/} = 0 
and therefore the characteristics satisfy either the equation 
(K + Pifh)8x + {hy + qJig)Sy = 0 . . . . (28) 
or else the equation 
{d^+p-^d^hx + {dy-\- q-^rt^hy = 0 .... (29) 
Substituting in these from t]ie first of equations (22), we see that the 
characteristics satisfy either the equation 
hxhx + hy^y + ..... (30) 
or else the equation 
d^h' + dyhy + dz^z=i) . . . . . (31) 
Moreover, from (26) the equation (28) may be written 
{ — dy p^h^hx + — 9, 
and substituting in this from the second equation of (22), it becomes 
- dy^x + d^^y + h^U = 0 ..... (32) 
Similarly, equation (29) leads to the equation 
hyhx-hJ>y + d^U (33) 
Thus the characteristics satisfy either the pair of equations (30) and (32), 
or else the pair of equations (31) and (33). But the equations (30) and (32) 
are the equations which define the magnetopotential surfaces, and the 
equations (31) and (33) are the equations which define the electropotential 
surfaces. Thus finally we have the result that the characteristics, out of 
which the calamoids are constituted, are all the curves that can he drawn 
arbitrarily on the electropotential surfaces, together with all the curves that 
can he drawn arbitrarily on the magnetopotential surfaces. This result 
might indeed have been foreseen from the mode in which the calamoids 
were formed originally : for any curve on an electropotential surface S 
may be regarded as the intersection of S with a continuous set H of oo ^ 
magnetopotential surfaces : and by § 7 this intersection forms part of every 
one of the calamoids which can be obtained as the intersection of H with a 
continuous set of oo ^ electropotential surfaces of which S is one : the curve 
is therefore an intersection of calamoids, that is, a characteristic. 
§ 14. Extension to General Fields. 
Throughout the whole of the preceding discussion we have been subject 
to one restrictive condition, namely, that the field is one in which the 
electric and magnetic vectors are everywhere perpendicular to each other. 
We have now to show what modifications are necessary when this condition 
is no longer supposed to hold. 
