49 
of Edinburgh, Session 1878-79. 
[In practice, the dotted line may conveniently be drawn with a 
coloured pencil or crayon.] 
A little consideration shows that — - 
1. Of the four angles at each crossing, one is enclosed between 
full lines, and its vertical angle by dotted [coloured] lines. These 
will be called the symmetrical angles. 
2. The crossing is electrodynamically positive if it is over to the 
right in the symmetrical angles, and vice versa. 
[In the figure the two interior crossings alone are positive.] 
3. If the knot be cut through along a line dividing the sym- 
metrical angles at any crossing, and the pairs of ends on either side 
of that line be reunited, the whole remains a knot, with one 
crossing less than before (Proc., 1877, p. 322). If the line divide 
the unsymmetrical angles, the whole becomes a link. 
[Dividing the figure at the upper crossing, it becomes either the 
twist of five-fold knottiness, or the trefoil knot once linked with a 
simple ring.] 
These methods are in practice very much superior in convenience 
of application to those I have already given, especially when the 
knot to be reduced is complex. 
The paper contains rules for the calculation of the beknottedness 
of the original knot, in terms of the beknottedness and belinkedness 
of these reduced forms ; so that knottiness n is made to depend 
upon n - 1. I have not yet succeeded in obtaining from these a 
general expression such as will take account of all the successive 
reductions of a knot to zero of knottiness. 
3. Preliminary Note on the Measurement of the Thomson 
Effect by the Aid of Currents from the Gramme Machine. 
By Professor Tait. 
