370 Mr. 0. J. Lodge on Nodes and Loops 



on the plane will be, that in the latter the bonds are not usually 

 made to cross each other, whereas in the former they will be 

 very liable to do so. The benzol chain might be projected out 

 of space as in fig. 2 ; and this form, though probably 

 less true, and certainly less convenient than the usual ^S- 2 « 

 form, appears to offer certain advantages. For if it be e<j^ a 

 considered experimentally established that the atoms | ^J<^ j 

 2 and 6 are similarly related to 1, then the fact is 5 \]x 3 

 plainly exhibited in the diagram. Moreover it will be 

 observed that 2, 4, and 6 are directly united to 1, while 3 and 5 

 are only indirectly connected ; and I believe that 3 and 5 are con- 

 sidered to possess, affinities distinctly differing from the others 

 when 1 is combined, say, with hydroxy], as in phenol. 



In counting the loops of the above configuration (fig. 2), it 

 must be noticed that 3 crossings and a loop run together at the 

 middle. The apparent loops are therefore 7, but 7 — 3 = 4 is 

 the real number. 



(4) To investigate the whole subject from a general point of 

 view I shall make use of the fact pointed out above, that when- 

 ever two bonds cross, a spurious loop is introduced — but shall 

 no longer consider a crossing as accidental, and its loop as 

 spurious. State the rule thus : — Every node added to a linkage* 

 made of a fixed number of rods, adds a loop. 



Take a loop of string and throw it flat on the table in a 

 tangle ; every node or crossing point accounts for a loop, and 

 there was one loop to begin with, therefore for n nodes there 

 are n-{- 1 loops. Now cut the string in any place, one loop 

 is destroyed, and it becomes a single open curve; cut it 

 again in another place, it is now two such curves, and another 

 loop is lost. And; generally, if it be made into 

 p such curves p loops will be lost, the number ^S* 3. 

 remaining being n— p + 1. Or \i p open curves 

 be drawn crossing each other or themselves at ran- 

 dom on a plane, the number of nodes exceeds the 

 number of loops by p — 1. In fig. 3 p = 7. 



A caution is here necessary: when cutting the string, bits must 

 not be cut out and left to lie wholly detached and not crossing any 

 thing ; for such a piece, though lying close to or inside the tangle, 

 has no more connexion with it than if it lay a mile away. 



A jumble of p rods, crossed as above, may be considered 

 as a casement without a frame. If a simple frame of any shape 

 be fitted to it so that both ends of every rod terminate in the 

 frame, the number of nodes remains n, but 2p fresh loops are 

 added. The whole number of panes in any casement is therefore 



* I do not quite know how far it is right to apply this term of Professor 

 Sylvester's to statical combinations of rods having no reference to motion. 





