
PROFESSOR TAIT ON KNOTS. 185, 

foe a 
DEE CA Deen ELA. 
ee 
MDE ADO! 

—+—+-+-4-+ 
The lower signs refer to over or under, the upper to the electro-magnetic work, 
or to the silver-copper distinction. 
Hence to determine the electro-magnetic work we must divide each scheme 
into independent circuits, no one of which includes a less extensive one; and 
omit from the reckoning the work for the terminal of each such circuit, and for 
each of the intersections which is not included in any one of the separate circuits. 
There are usually more ways than one of doing this. Sometimes these agree 
in their results ; but the rule for choosing which to omit is to take them such 
that with their proper signs, and the rest with any signs whatever, they may 
be capable of making each letter positive on its first appearance. But there 
are cases even when the knot is not amphicheiral in which this process cannot 
be carried out. These occur specially when a part of the knot forms a lowe1 
knot with which the string is again linked. 
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 ; 2¢., itis — 167. 
But in the second scheme we may take the two circuits 
BADBandCADC, 
and E is not included in either. Hence we must leave out of count the work 
for B, C, and E. This is found to satisfy our test, and thus the whole work 
is only — 8 zw. 
This is an instance in which the estimate by the electro-magnetic process 
exactly agrees with the result of simpler considerations, as given in § 35 above. 
§ 44. It will be found that the alteration of five signs is sufficient to remove 
the knotting from the annexed figure, and the stages of operation of the various | 
modes of reduction show that this form can be regarded as 
made up of simpler knots intersecting one another on the same 
strmg. ‘These separate knots are virtually independent, and to 
change ail the signs in any one of them does not in cases like 
this necessarily simplify the knot. Uncorrected the work is 
— 13x 4a. Corrected it is— 10 x 47, which agrees with the removal of the 
beknottedness by change of jive signs only. 
VOL. XXVIII. PART I. 3B 
