text: In chemical kinetics, the elementary reactions are represented by the stoichiometric equations
  where  are the components and  are the stoichiometric coefficients. Here, the reverse reactions with positive constants are included in the list separately. We need this separation of direct and reverse reactions to apply later the general formalism to the systems with some irreversible reactions. The system of stoichiometric equations of elementary reactions is the reaction mechanism.
  The stoichiometric matrix is ,  (gain minus loss). This matrix need not be square. The stoichiometric vector  is the rth row of  with coordinates .
  According to the generalized mass action law, the reaction rate for an elementary reaction is
  where  is the activity (the "effective concentration") of .
  The reaction mechanism includes reactions with the reaction rate constants . For each r the following notations are used: ; ;  is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not;  is the reaction rate for the reverse reaction if it is in the reaction mechanism and 0 if it is not. For a reversible reaction,  is the equilibrium constant.
  The principle of detailed balance for the generalized mass action law is: For given values  there exists a positive equilibrium  that satisfies detailed balance, that is, . This means that the system of linear detailed balance equations
  is solvable (). The following classical result gives the necessary and sufficient conditions for the existence of a positive equilibrium  with  detailed balance (see, for example, the textbook).
  Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:
  # If  then  and, conversely, if  then  (reversibility);
  # For any solution  of the system 
  the Wegscheider's identity holds:
  Remark. It is sufficient to use in the Wegscheider conditions a basis of solutions of the system .
  In particular, for any cycle in the monomolecular (linear) reactions the product of the reaction rate constants in the clockwise direction is equal to the product of the reaction rate constants in the counterclockwise direction. The same condition is valid for the reversible Markov processes (it is equivalent to the "no net flow" condition).
  A simple nonlinear example gives us a linear cycle supplemented by one nonlinear step: 
  There are two nontrivial independent Wegscheider's identities for this system:
   and 
  They correspond to the following linear relations between the stoichiometric vectors:
   and 
  The computational aspect of the Wegscheider conditions was studied by D. Colquhoun with co-authors.
  The Wegscheider conditions demonstrate that whereas the principle of detailed balance states a local property of equilibrium, it implies the relations between the kinetic constants that are valid for all states far from equilibrium. This is possible because a kinetic law is known and relations between the rates of the elementary processes at equilibrium can be transformed into relations between kinetic constants which are used globally. For the Wegscheider conditions this kinetic law is the law of mass action (or the generalized law of mass action).