58 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



second of the above forms of the condition of equilibrium, which will 

 then become / \ > n / Q x 



(0 e M V, etc. = 0, (9) 



rf, ty", etc., denoting the entropies of the various parts between which 

 there is no communication of heat. When the condition of equi- 

 librium is thus expressed, the limitation in respect to the conduction 

 of heat will need no farther consideration.-' 



In order to apply to any system the criteria of equilibrium which 

 have been given, a knowledge is requisite of its passive forces or 

 resistances to change, in so far, at least, as they are capable of pre- 

 venting change. (Those passive forces which only retard change, 

 like viscosity, need not be considered.) Such properties of a system 

 are in general easily recognized upon the most superficial knowledge 

 of its nature. As examples, we may instance the passive force of 

 friction which prevents sliding when two surfaces of solids are 

 pressed together, that which prevents the different components of 

 a solid, and sometimes of a fluid, from having different motions one 

 from another, that resistance to change which sometimes prevents 

 either of two forms of the same substance (simple or compound), 

 which are capable of existing, from passing into the other, that 

 which prevents the changes in solids which imply plasticity, (in other 

 words, changes of the form to which the solid tends to return,) when 

 the deformation does not exceed certain limits. 



It is a characteristic of all these passive resistances that they pre- 

 vent a certain kind of motion or change, however the initial state of 

 the system may be modified, and to whatever external agencies of force 

 and heat it may be subjected, within limits, it may be, but yet within 

 limits which allow finite variations in the values of all the quanti- 

 ties which express the initial state of the system or the mechanical 

 or thermal influences acting on it, without producing the change in 

 question. The equilibrium which is due to such passive properties 

 is thus widely distinguished from that caused by the balance of the 

 active tendencies of the system, where an external influence, or a 

 change in the initial state, infinitesimal in amount, is sufficient to pro- 

 duce change either in the positive or negative direction. Hence the 

 ease with which these passive resistances are recognized. Only in 

 the case that the state of the system lies so near the limit at which 

 the resistances cease to be operative to prevent change, as to create a 

 doubt whether the case falls within or without the limit, will a more 

 accurate knowledge of these resistances be necessary. 



To establish the validity of the criterion of equilibrium, we will 

 consider first the sufficiency, and afterwards the necessity, of the con- 

 dition as expressed in either of the two equivalent forms. 



In the first place, if the system is in a state in which its entropy is 



