in connexion with Chemical Formulae. 375 



is that in the equation of a chemical reaction where a compounds 

 on one side are equated to b compounds on the other, and where 

 x is the sum of the number of loops of the a compounds, there 

 the sum of the number of loops of the b compounds is 



x — a + b. 



With this result the number of loops in any compound may be 

 immediately found by equating it to all its elements ; and this 

 indeed is the most direct way of establishing the general formula 

 of § 8. 



(13) We found in § 11 that every detached compound corre- 

 sponds to a negative loop ; hence a single compound (being de- 

 tached) ought strictly to be considered as giving a negative loop; 

 or the real number of loops in a compound is in all cases one less 

 than the number which we have so far been satisfied with, and 

 which all our expressions give. 



r(k—2) 

 Thus in a linkage of r atoms of a #-ad there are ~*«-g — - + 1 



loops ; therefore in no atoms of an any-ad there is 1 loop — which 

 would have seemed absurd, only the reason is now plain, viz. that 

 the presence of a compound is itself a negative loop, and so its 

 removal adds a loop. The presence of the 1 in all our formulae 

 is thus explained. Taking the negative loop of presence into 

 account, it disappears; and then no atoms of a #-ad give no loops. 

 Further, there is no reason why we should limit ourselves to 

 atoms of positive atomicity; in fact we have not done so ; for we 

 shall see that monads are really negative. Thus, take a homo- 

 geneous linkage of £-ads, represent number of loops as ordinate?, 

 and number of atoms as abscissae ; the curve is a straight line 

 for every value of k, but it has different slopes. It makes 45° 

 with the axes for 7^ = 4 or tetrads, and again 45° in the other 

 quadrants for k = Q or no-ads, while for k = 2 the line is parallel 

 to the axis of x. Hence 2 is the natural zero of atomicity (at 

 any rate in " loop "-considerations), not 0. If this natural zero 

 were used instead of the ordinary one, and if further the " pre- 

 sence-loop n were taken into account, all the expressions would 

 simplify and become more symmetrical. Every atom would 

 then give a number of loops equal to half its natural atomicity, 

 and the general formula of § 8 would become 



Thus, then, monads have negative bonds, and subtract loops 

 instead of adding them . Whatever a (2 + k) -ad does, a (2 — k) -ad 

 can undo ; e, g. every tetrad adds a loop, every no-ad subtracts 

 one, and so on. 



Moreover there is no necessity for limiting our statements to 



