446 Prof. Cayley on the Mathematical Theory of Isomers. 



I V i 





i * II 



* 



(-) 08) (7) 



(«) (Pi (7) (8) 



where the mathematical question of the determination of such 

 forms belongs to the class of questions considered in my paper 

 " On the Theory of the Analytical Forms called Trees," Phil. 

 Mag. vol. xiii. (1857) and vol. xv. (1859), and in some papers 

 on Partitions in the same Journal. 



The different forms of univalent diagrams Q n H 2w+1 are ob- 

 tained from the same ramifications by adding to each of them 

 all but one of the 2/1 + 2 points H ; that is, by adding to each 

 point C except one its full number of points H, and to the ex- 

 cepted point one less than the full number of points H. The 

 excepted point C must therefore be univalent at least ; viz. it 

 cannot be a saturate point, which presents itself for example in 

 the diagrams 71 = 5(7) an( l ra=6(S). And in order to count the 

 number of distinct forms (for the diagrams C n H 2n+1 ), we must 

 in each of the above ramifications consider what is the number 

 of distinct classes into which the points group themselves, or, say, 

 the number of " allotrious " points. For instance, in the rami- 

 fication tz=3 there are two classes only; viz. a point is either 

 terminal or medial ; or, say, the number of allotrious points is 

 = 2: this is shown in the diagrams by means of the asterisks; 

 so that in each case the points which may be considered allo- 

 trious are represented by asterisks, and the number of asterisks 

 is equal to the number of allotrious points. 



Thus, number of univalent diagrams C n H 2 " +1 : — 



»=1, 1 



n = 2, 1 



n = 3, 2 



7i=4, (a)2;(/S)2; together 4 



n = 5, («) 3; (0)4; ( y ) 1 ; . . „ 8 



n = 6, («)3; 08)5; (7) 2; (5)3; „ 18 



