700 
MR. A. McAULAY ON THE MATHEMATICAL 
25. The differentiations with regard to time implied by these dots are differentia¬ 
tions for a fixed element of the standard position of matter, i.e., they are differentiations 
that follow the motion of matter. It is clear then that they are commutative with V, 
hut not with V'. Hence, from equations (7) and (10), 
SVC = 0, [SdZG] a + 0 = 0.(LI), 
and, therefore, [equation (8) § 8 above] 
SV'C'= 0, [SdtV] a+6 = 0.. . (12). 
Since C satisfies the conditions of incompressibility, its surface integral over any 
surface only depends on the boundary of the surface, and may be expressed as the line 
integral of a vector K/47T round it. Thus, by equation (3), § 5, 
4ttC = VVH.(13). 
H is called the magnetic force, and is assumed to be an intensity, so that (Prop. III., 
§ 8 ) 
47rC' = VV'H'.(14). 
All the vectors, including II, hitherto mentioned, may be discontinuous. But they 
are assumed to be finite, so that JjjCdy = 0 for any infinitely small volume. Suppose 
this volume is a disc enclosing a part of a surface of discontinuity in H. Then we 
have 
0 = jjjvVHefc = jjVcZSH 
by equation (4), §5 above. Hence 
[VcfcH]„ + j=0.(15), 
so that the discontinuity in II is entirely normal to the surface. Similarly 
[Ydt lTWa=0.(16). 
26. From what has been said it follows that if d, k and their rates of variation are 
given for every point of space, H is not yet completely determined. It is, however, so 
determined by one more condition which is proved in § 48 below, and which is given 
here as we shall want to use it before proving it. H is one of the independent 
variables of which l is supposed an explicit function. The condition mentioned is that 
H VZ satisfies the conditions of incompressibility. In other words, putting 
