374 Mr. O. J. Lodge on Nodes and Loops 



Jc—2 bonds to each, and joining the bonds in pairs; every pan* 

 of bonds adds a loop to the figure, and there was the original 

 ring to start with. 



Since the number of loops in a compound is determined by 

 its empirical formula, or number of atoms, it is evidently inde- 

 pendent of the arrangement of those atoms ; consequently all iso- 

 mers have the same number of loops in their composition. Also if 

 any graphic formula has no loops in it, it cannot be made to have 

 any by any rearrangement of the atoms. But it must be under- 

 stood that the loops formed by M latent" bonds are to be counted ; 

 otherwise one kind of ethylic cyanide, for instance, would have 

 two loops more than the other. 



(11) Several of the formula? are apt to give a negative num- 

 ber of loops in certain cases. There is a certain amount of 

 meaning in this. For instance, a single monad molecule has 

 no loops and no nodes; two detached monad molecules must 

 be considered as having no nodes and — 1 loop ; if -p. « 

 their bonds are laid across each other you get 1 node; v/ 

 and (§ 3) the node introduces a loop, raising the I ] /\ 

 number of loops to (v. fig. 7). If one atom of 



each molecule in the crossing pair be made a dyad, a loop is 

 formed; not so with the detached pair. It is true that if 

 both the atoms of a single molecule be made dyads a loop is 

 obtained ; but the detached molecule must be considered as sub- 

 tracting a loop, leaving as the total number. In fact with n 

 detached simple molecules the number of loops is 1 — n. Three 

 detached hydrogen molecules have — 2 loops; to make one loop 

 with them you have to cross the bonds three times. Of course 

 the real number of loops is not changed by any amount of cross- 

 ing, those introduced by the nodes are what I formerly (§ 3) 

 called spurious ones. The number of nodes appears incapable 

 of being negative, except under the imaginary circumstances of 

 §13. 



(12) It will now be seen that detached rods or rings (§ 4) may 

 be considered as having some connexion with our entanglements, 

 and may be taken into consideration by means of the convention 

 just stated ; but a former statement remains true, that the con- 

 nexion does not vary with the distance. 



Water of crystallization may be said to lie in detached simple 

 molecules ; and if the number of loops in a crystallized com- 

 pound be determined from its complete formula instead of from 

 that of the anhydrous salt, the number obtained will not be erro- 

 neous if one is willing to consider that every detached water 

 molecule corresponds to a negative loop. The same is true of 

 any kind of molecular combination other than that usually repre- 

 sented by bonds, as in double salts. A further consequence of § 11 



