






186 PROFESSOR TAIT ON KNOTS. 
If the sign of the one unsymmetrical crossing be altered, four changes of 
sign will suffice; for the uncorrected work is —11 x 47; corrected it is 
— 8 x 47, corresponding to four changes of sign. 
§ 45. It is clear from what precedes that the GAvsSIAN integral does not, 
except in certain classes of cases, express the measure of what may be called, 
by analogy with § 35, Belinkedness. It may be well to examine a simple form 
of link with all its possible arrangements of sign to see what the integral really 
gives in each of these. Let us choose for this purpose two lemniscates having 
four mutual crossings, as in the edges of the band shown in fig. 13, Plate XV. 
If we suppose the signs to be made alternately + and —, as in Plate XVI. 
fig. 10, the form is a six crossing one, and irreducible. The silver or copper 
character of the se/f crossings does not depend upon the directions in which we 
suppose the lemniscates to be described, that of the mutual crossings does. We 
thus have, from another point of view than that of § 41, a proof that these are 
to be left out of account in the reckoning. 
The four crossings of the two curves are copper, if these curves are supposed 
to be described in the same way round ; those of the separate curves (which 
do not count) are silver. Hence the work is —16 z, or two degrees of belinked- 
ness. 
Change the sign of any one of P, Q, R, S, that and the adjacent one slip 
off, U and V become nugatory. The linkage is the simplest possible, and the 
integral is 8 7. 
Change the sign of either or both of U and V. In either of these three 
cases both become nugatory, and the whole takes the form of two doubly- 
linked ovals, with the integral = — 167. (Plate XVI. figs. 12, 13.) 
If the signs of both R and S be changed the value of the integral is 
obviously 4 (2—2) a, for R and S have become silver, while P and Q remain 
copper. 
If in addition the signs of U and V be both, or neither, changed, only one 
crossing is got rid of, and the link may be put in the form (Plate XVI. fig. 14). 
It cannot be farther reduced, because the crossings are alternately over and 
under. 
But if the sign of one only of U, V be changed, it will be seen that there is 
no linking (Plate XVI. fig. 11). Here the integral vanishes because there is 
really no work, not as in the last case, where there are equal amounts of positive 
and negative work. 
§ 46. This gives a hint as to the reckoning of beknottedness from the silver 
and copper crossings in the cases where we have found a difficulty. After 
omitting from the reckoning the crossings which belong merely to the outline 
of the figure, there must remain an even number of crossings (§ 22). Hence, 
whatever numbers be silver and copper respectively, the excess of the one of 
