293 
of Edinburgh, Session 1876-77. 
of my paper, which, for the comparison below, must be supposed to 
have all its signs changed) whose schemes are, respectively — 
APBECAPBEC 
— + — + — -b — + — + 
A, 
ADBECADCEB 
- + - + -+•- + - + 
A. 
The lower signs refer to over or under, the upper to the electro- 
magnetic work, or to the silver-copper distinction. These two 
instances, in which both series of signs are absolutely identical, 
each with each, show at once that we cannot take the two sets of 
signs alone as fully descriptive of the knot. 
To determine the electromagnetic work for any knot, we must 
divide the scheme into independent circuits, no one of which in- 
cludes a less extensive one ; and omit from the reckoning the work 
for the terminal of each such circuit, and for each of the intersec- 
tions which is not included in any one of the feparate circuits. The 
particular closed circuits chosen do not affect the final result, as is 
easily seen by thinking of the various deformations of each figure. 
In the first of the two schemes above there is but one independent 
non-autotomic circuit, which may be taken as 
ADBECA. 
In this all the intersections are included, so that the whole work is 
to be found by leaving out that for A only ; i.e., it is - IGtt. 
But in the second scheme we may take the two circuits 
BADBand CADC, 
and E is not included in either. Hence we must leave out of count 
the work for B, C, and E ; and thus the whole work is only 
— 87T. 
In fact, the figures show that to untie the first knot we must not 
only have the signs such that we can slip off B and E, but also C 
and P, i.e., two signs must be changed; while the second loses all 
its beknottedness if A and P could be got rid of, that is if one sign 
only be changed. This is an instance in which the estimate by the 
electro-magnetic process exactly agrees with the result of simpler 
