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



(6) From an wz-node 2m lines radiate; but an odd number of 

 lines may radiate from those nodes in which a rod terminates 

 without going through. Call these " odd nodes." The smallest 

 number of rods required for the construction of any jumble will 

 be half the whole number of odd nodes in it ; for a rod can always 

 pass completely through an even node, but must ultimately stop 

 at an odd node, even though it go through it to begin with. 



From this it follows that, since any arrangement of closed 

 curves will have only even nodes, such an arrangement may 

 always be drawn without taking the pen off. The same may be 

 done if a diagram contains two odd nodes; but you must begin 

 at one of them and will leave off at the other. The Fig. 5. 

 child's puzzle, fig. 5, since it contains 4 odd nodes, 

 requires 2 rods to make it, and so cannot be drawn fT\7T\ 

 without taking the pen off. A Lissajous's figure, ^\Z^J 

 being a reentrant curve, has always one loop in ex- 

 cess of the nodes. There is an even number of odd nodes in every 

 tangle. 



(7) I shall now represent graphic formulae by arrangements 

 of rods. An ordinary node will represent a tetrad atom, and a 

 node from which k lines radiate, a #-ad atom ; odd nodes will 

 be called perissads, and even nodes artiads. Every compound 

 will be supposed to be made of the minimum number of rods, 

 viz. half the number of its perissad atoms. Examples of 

 these formulae are given in fig. 6, which represents the com- 

 pounds carbonic anhydride, ethylene, nitric acid, calcic oxalate, 

 benzol, acetic acid, and ferrocyanide of potassium. I con- 

 sider that such skeleton formulae may be useful in general inves- 

 tigations, where the atomicity of an element is the principal 

 property to be attended to. They have the advantage of being 



Fig. 6. 



easy to draw. In fig. 6 the HN0 3 has 1 rod, 2 loops, and 2 

 nodes ; the K 4 FeCy 6 has 5 rods. Common alum has only 1 rod, 

 10 nodes, and 10 loops. A paraffin requires n + 1 rods. 



The nodes in these figures are not to be confounded with a 

 crossing of the bonds ; in them, as in other graphic formula?, 

 the bonds do not cross. An atom exists at every node; and its 

 atomicity, which is exhibited, is in general sufficient to give its 

 name. Dyad atoms are like singular points or knots on the 

 string of § 4. (Professor Cayley's tree " knots >J are tetrad 

 atoms.) There is no necessity for dotting in the hydrogen 

 atoms ; free bonds are best represented by arrow-heads. 



(8) We have proved (§§4 and 6) that, calling Pthe number 



