in connexion with Chemical Formula. 371 



w-J-jo + 1 (fig. 4). If the connexions with the frame Fig. 4. 

 are counted as nodes, it belongs to the former case. 



If the free ends of the jumble of p rods be joined 

 in pairs, p fresh loops are formed, the figure becomes 

 a set of closed rings crossing each other, and 

 the number of its loops (being ft — p-\-\+p) exceeds the 

 number of crossings by 1. Here also no ring must lie without 

 crossing any thing ; every such detached ring causes one loop 

 too many. It may in fact be said to give only one imaginary 

 loop for two imaginary crossings, the other loop (viz. itself) being 

 real. 



(5) The maximum number of nodes possible with m straight 



rods is — —= — \ This is evident by constructing a jumble, rod 



by rod : the second rod gives one node, the third gives two more, 

 and the ?wth gives (m — 1) more. Now in any of the above en- 

 tanglements several lines are apt to cross at one and the same 

 point ; when this happens some loops and nodes are lost. How 

 many? 



Let m lines cross at a single point (such a point may be called 

 an w-node), the actual number of crossings which there run 

 together is evidently the maximum number of nodes possible 



with m straight rods, viz. — ~. The number of loops in 



the point will be, by the general rule, m — 1 less or ~ -. 



But the effect of such m-nodes in a tangle may be most readily 

 allowed for by saying that each is equivalent to m-1 ordinary 

 nodes, the remaining nodes being compensated by vanishing 

 loops. This is true whether the rods run through the point or 

 only run into it and stop ; so, if one rod runs into another and 

 stops without crossing it, it must count as an ordinary node. If 

 a rod terminates in an ra-node it adds to it one node more. The 

 touching point of two curves must be counted as a single node, 

 no matter how high the order of contact may be. All these things 

 have followed from the general law thatjo rods give^ — 1 more 

 nodes than loops ; so that in any figure without loops the num- 

 ber of nodes must be p — 1. To any jumble ofp open curves or 

 " rods " any number of closed rings may be added, and the law 

 of loops and nodes will remain true : but the closed curves must 

 not, of course, be counted among the^> open ones ; they may be 

 entirely neglected. 



It can be easily proved that, if the assemblage of ellipses passing 

 from a straight line through a circle to a straight line cutting 

 the first at right angles can be inscribed in a square and havea? 

 ■loops, the number 2x+l is a perfect square. 



3B2 



