





180 PROFESSOR TAIT ON KNOTS. 
sides of the uniformly and normally magnetized surface whose edge is the 
circuit. There is, however, no discontinuity in the value of the work, for the 
element of the double integral is finite, and equal to 47, when Ta=0. | 
Gauss * says (with date January 22, 1833) :—“ Eine Hauptaufgabe aus dem 
Grenzgebiet der Geometria Situs und der Geometria Magnitudinis wird die sein, 
die Umschlingungen zweier geschlossener oder unendlicher Linien zu zahlen.” 
And he adds that the integral 

y (x’ —x)(dydz — dada’) + (y'—y)(dada' — dadz) + (z —2)(dady’ — dydz’) ; 
(@—a)+ Y—9)? + @—2)? )s 
extended over both curves, has the value 
4m, 
where m is the number of linkings (Umschlingungen). This is obviously the 
same as the integral of 6W above, viz. :— 
a, S.adaéa 
Tae > 
extended round each of two closed curves, of which da and éa are elements. 
§ 39. A very excellent investigation, by means of Cartesian co-ordinates, 
will be found in CLERK-MAxweEL1’s Electricity and Magnetism §§ 417-422. It is 
there shown that the above integral may vanish, even when the circuits are 
inseparably linked together. In fact m may vanish either because there is no 
real linking at all, or because the number of linkings for which the electro- 
magnetic work is negative is the same as that for which it is positive. For 
our present application this is of very great consequence, because it shows that 
the electro-magnetic work, under the circumstances with which we are dealing, 
cannot in all cases measure the amount of beknottedness. In fact the processes, 
soon to be described, enable us, without trouble for any given linkage, to find the 
value of m in Gauss’ formula; but there are special ambiguities when we try to 
apply the process to knots. 
§ 40. To construct the magnetized surface which shall exert the same action 
on a pole as a current in any given closed circuit does, we may 
either suppose a surface extending to infinity in one direction 
(say for definiteness, upwards from the plane of the paper), and 
having the circuit for its edge; or we may form, as in the figure, 
a finite autotomic surface of one sheet, having the circuit for 
its edge. In dealing with the two curves of. GAUSS’ proposi- 
tion, our procedure is perfectly definite ; but when one and the same curve is 
to be the current and also the path of the pole, there is an ambiguity in esti- 


* Werke, Gottingen, 1867, v. p. 605. 
