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VIII.—On Knots. By Professor Tarr. - (Plates XV. and XVI.) 
(Revised May 11, 1877.) 
The following paper contains, in a compact form, the substance of several 
somewhat bulky communications laid before the Society during the present 
session. The gist of each of these separate papers will be easily seen from the 
abstracts given in the Proceedings. These contain, in fact, many things which 
I have not reproduced in this digest. Nothing of any importance has been - 
added since the papers were read, but the contents have been very much 
simplified by the adoption of a different order of arrangement ; and long passages 
of the earlier papers have been displaced in favour of short general statements 
from the later ones. With the exception of the portion which deals with the 
main. question raised, this paper is fragmentary in the extreme. Want of 
leisure or press of other work may justly be pleaded as one cause; but there is 
more than that. The subject is a very much more difficult and intricate one than 
at first sight one is inclined to think, and I feel that I have not succeeded in 
catching the key-note. When that is found, the various results here given will 
no doubt appear in their real connection with one another, perhaps even as 
immediate consequences of a thoroughly adequate conception of the question. 
I was led to the consideration of the forms of knots by Sir W. THomson’s 
Theory of Vortex Atoms, and consequently the point of view which, at least at 
first, 1 adopted was that of classifying knots by the number of their crossings ; 
or, what comes to the same thing, the investigation of the essentially different 
modes of joining points in a plane, so as to form single closed ee curves with a 
given number. of double points. 
The enormous numbers of lines in the spectra of certain dgnewnag sub- 
stances show that, if THomson’s suggestion be correct, the form of the corre- 
‘sponding vortex atoms cannot be regarded as very simple. For though there 
is, of course, an infinite number of possible modes of vibration for every vortex, 
the number of modes whose period is within a few octaves of the fundamental 
mode is small unless the form of the atom be very complex. Hence the diffi- 
culty, which may be stated as follows (assuming, of course, that the visible rays 
emitted by a vortex atom belong to the graver periods) :—“ What has become 
of all the simpler vortex atoms ?” or “ Why have we not a much greater number 
of elements than those already known to us ?” It will be allowed that, from 
the point of view of the vortex-atom theory, this is almost a vital question. 
Two considerations help us to an answer. First, however many simpler 
forms may be geometrically possible, only a very few of these may be forms of 
VOL, XXVIII. PART I. 2 P 
