The collected mathematical papers of Arthur Cayley.

610]

427

610.
ON THE ANALYTICAL FORMS CALLED TREES, WITH APPLICATION TO THE THEORY OF CHEMICAL COMBINATIONS.
[From the Report of the British Association for the Advancement of Science, (1875),
pp. 257-305.]
I HAVE in two papers "On the Analytical forms called Trees," Phil. Mag. vol.
xIII. (1857), pp. 172-176, [203], and ditto, vol. xx. (1859), pp. 374-378, [247], considered this theory; and in a paper "On the Mathematical Theory of Isomers," ditto,
vol. XLVII. (1874), p. 444, [586], pointed out its connexion with modern chemical theory.
In particular, as regards the paraffins CnH,+2, we have n atoms of carbon connected
by n -1 bands, under the restriction that from each carbon-atom there proceed at
most 4 bands (or, in the language of the papers first referred to, we have n knots
connected by n-1 branches), in the form of a tree; for instance, n =5, such forms
(and the only such forms) are
3 3 3 3
h 1 2' ' 3 — o 3
2
2< 3
63
36
And if, under the foregoing restriction of only 4 bands from a carbon-atom, we
connect with each carbon-atom the greatest possible number of hydrogen-atoms, as
shown in the diagrams by the affixed numerals, we see that the number of hydrogenatoms is 12 (=2. 5 + 2); and we have thus the representations of three different
paraffins, CH12. It should be observed that the tree-symbol of the paraffin is
54-2