258 



REPORT 1875. 



And if (under tlie 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 hydrogen-atoms is 12 (=2'5-(-2), and we have thus the repre- 

 sentations of three different paraffines, C. H^^. It should be observed that 

 the tree-symbol of the parafflne is completely determined by means of the 

 tree formed with the carbon-atoms, or say of the carbon-tree, and that the 

 question of the determination of the theoretic number of the paraffines 

 C„ H2„_|.2 is consequently that of the determination of the number of the car- 

 bon-trees of n knots, viz. the number of trees with n knots, subject to the 

 condition that the number of branches from each knot is at most =4. 



In the paper of 1857 (which contains no application to chemical theory) 

 the number of branches from a knot was unlimited ; and moreover the trees 

 were considered as issuing each from one knot taken as a root, so that, n=5, 

 the trees regarded as distinct (instead of being as above only 3) were in all 

 9, viz. these were 



w 



which, regarded as issiiiug from the bottom knots, are in fact distinct ; whQe 

 taking them as issuing each from a properly selected knot, they resolve 

 themselves into the above-mentioned 3 forms. The problem considered 

 was in fact that of the " general root-trees with n knots "—(/eneral, inas- 

 much as the number of branches from a knot was without limit ; root-trees, 

 inasmuch as the enumeration was made on the principle last referred to. It 

 was found that for 



knots 1 2 3 4 5 G 7 8 



No. of trees was.. 1 1 2 4 9 20 48 115 



= 1 A, A., A, A, A, A„ A, . . . . 



the law being given by the equation 



but the next following numbers A^, A^, A,;,, the correct values of which are 

 286, 719, 1842, were given erroneously as 306, 775, 2009. I have since I 

 calciilated two more terms, A^j, A,^ = 4766, 12486. ■ 



The other questions considered in the paper of 1857 and in that of 1860 

 have less immediate connexion with the present paper, but for completeness 

 I reproduce the results in a Note*. 



* In the paper of 1857 I also considered the problem of finding B^ the number with r 

 free branches, bifurcations at least : this was given by a like formula — 



