EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 203 



between the quantities denoted by e v /, fly, T-->, . . . -T-?, involving also 



the quantities which express the composition of the body, when that 

 is capable of continuous variation, or any other equation from which 

 the same relations may be deduced, may be called a fundamental 

 equation for that kind of solid. It will be observed that the sense in 

 which this term is here used, is entirely analogous to that in which we 

 have already applied the term to fluids and solids which are subject 

 only to hydrostatic pressure. 



When the fundamental equation between e V '> ^7v> -j-, >> j~? i s 



known, we may obtain by differentiation the values of t, X x >, . . . Z v 

 in terms of the former quantities, which will give eleven independent 

 relations between the twenty-one quantities 



dx dz v 



y/ ' ^ v/> dx" ' ' ' dz" x/ ' ' ' ' z ' } (415) 



which are all that exist, since ten of these quantities are independent. 

 All these equations may also involve variables which express the 

 composition of the body, when that is capable of continuous variation. 

 If we use the symbol t/*v to denote the value of \js (as defined on 

 page 89) for any element of a solid divided by the volume of the 

 element in the state of reference, we shall have 



\/r v , = e v ,-^ v ,. (416, x 



The equation (356) may be reduced to the form 



x ,6j ; ). (417) 



Therefore, if we know the value of \fs v in terms of the variables t 



(liCf (I Z 



-j,, . . . -T,, together with those which express the composition of the 



body, we may obtain by differentiation the values of rj v >, X x >, . . . Z z , 

 in terms of the same variables. This will make eleven independent 

 relations between the same quantities as before, except that we shall 

 have \/r v . instead of e v >. Or if we eliminate \Js v by means of equation 

 (416), we shall obtain eleven independent equations between the 

 quantities in (415) and those which express the composition of the 

 body. An equation, therefore, which determines the value of \/s v , 



/Y/Y* ft ft 



as a function of the quantities t, -* . . . -1-7, and the quantities which 



express the composition of the body when it is capable of continuous 

 variation, is a fundamental equation for the kind of solid to which it 

 relates. 



In the discussion of the conditions of equilibrium of a solid, we 

 might have started with the principle that it is necessary and sufficient 



