309 
of Edinburgh , Session 1876-77. 
following are his figures: the first and second belonging to the 
former of the two equivalent symbols, the third to the latter — 
He concludes this part of his Essay by saying that u these 
examples (confined as they are to single, closed lines), and the 
remarks made upon them, serve to show that the fundamental 
conception of twisting of lines is capable of being applied to the 
most complex space relations.” 
It may be added that these very elegant and important results 
are given as statements merely, without any hint of how they were 
arrived at, or how they may be extended. In fact brevity has 
been sedulously studied, for all that is given about knots forms a 
comparatively small part of the whole of Listing’s extremely valu- 
able, but too brief, Essay. 
The rest treats, rather more fully, the whole subjects of inver- 
sion and perversion, screws of various kinds, plaiting and twisting, 
(with their applications to vegetable spirals, &c.), the numbers of 
jines joining given sets of points, the extensions of the meaning of 
the word Area , &c., &c. 
The above abstract, which contains almost all of Listing’s remarks 
on knots, shows that he has long ago anticipated a very great deal 
of what I have lately sent to the Society. For myself, however, I 
may say that I have had to learn only two things (about knots) from 
Listing, viz. (1), a special case (which will be examined imme- 
diately), in which two forms are equivalent, though not transform- 
able into one another by the methods given in my first paper; and 
(2), to value more highly than I had hitherto done the method of 
classifying forms by the numbers of each kind of mesh, and the 
right-handed or left-handed character of each. 
My first paper, as sent to the Society, was essentially confined 
(as indeed its title indicates) to the results deducible from a special 
elementary theorem, — one of two which occurred to me long ago 
