292 
Proceedings of the Royal Society 
sideration of the magnetised surface above mentioned. Go round 
the curve, marking an arrow head after each crossing to show the 
direction in which you passed it. Then a junction like the fol- 
lowing gives + 2 tt at each time of crossing ; 
or, if we use the infinite surface spoken of 
above, it gives + 4nr for the upper branch, 
and nothing for the lower (which, on this 
supposition, does not pass through the magnetic sheet). Change 
the crossing from over to under , and these quantities change sign. 
The junction figured above would, in our first illustration, be a 
silver one. But a still simpler process is to go round putting a 
dot to the right after each crossing over , and vice versa. Silver 
crossings have two dots in one angle ; copper one in each of two 
vertical angles. 
Now, in order that our rule may be such as to give no work 
where there is no beknottedness, we must make the required ex- 
pression such as to vanish whenever all the intersections are 
nugatory. Those which are nugatory only in consequence of 
their signs are in pairs, silver and copper, and will take care of 
themselves, as we see by special examples like the following, in 
which the reversal of one of the directions simply reverses the 
signs. Hence the only part to correct for 
is that depending on the number of whole 
turns, and the sketch of the india-rubber 
band above shows that the work at the 
vertex of each such partial closed circuit is simply not to be 
counted-— i.e., that the ± 47r , which would be reckoned for each 
crossing by our rule, is to be considered as made up for by the cor- 
responding screwing of the pole round the curve. 
To illustrate the application of this process, let us consider again 
the two distinct forms with five non-nugatory intersections 
1 . 
(the first being a modified form of the “pentacle,” the second, fig. 6 
