THERMODYNAMICS OF FLUIDS. 13 



,.,.,, de I j a dv 



which is then reduced to -=~dn , 



e c c v 



and by integration to loge=- logv.* (D) 



c c 



The constant of integration becomes 0, if we call the entropy for 

 the state of which the volume and energy are both unity. 



Any other equations which subsist between v, p, t, e and r\ may be 

 derived from the three independent equations (A), (B) and (D). If we 

 eliminate e from (B) and (D), we have 



7/ = alog / y + clog^H-clogc. (E) 



Eliminating v from (A) and (E), we have 



tj = (a+c)\ogt alogp+clogc+aloga. (F) 



Eliminating t from (A) and (E), we have 



/ 



ij = (a+c)logv+clogp+c\og-. (a) 



ot 



If v is constant, equation (E) becomes 



T] = c log t + Const., 



i.e., the isometrics in the entropy-temperature diagram are logarithmic 

 curves identical with one another in form, a change in the value of 

 v having only the effect of moving the curve parallel to the axis of tj. 

 If p is constant, equation (F) becomes 



T] = (a + c) log t + Const., 



so that the isopiestics in this diagram have similar properties. This 

 identity in form diminishes greatly the labour of drawing any con- 

 siderable number of these curves. For if a card or thin board be cut 

 in the form of one of them, it may be used as a pattern or ruler to 

 draw all of the same system. 



The isodynamics are straight in this diagram (eq. B). 



To find the form of the isothermals and isentropics in the volume- 

 pressure diagram, we may make t and r\ constant in equations (A) 

 and (G) respectively, which will then reduce to the well-known equa- 

 tions of these curves : 



pv Const., 



and c v a+c Const. 



*If we use the letter to denote the base of the Naperian system of logarithms, 

 equation (D) may also be written in the form 



This may be regarded as the fundamental thermodynamic equation of an ideal gas. See 

 the last note on page 2. It will be observed, that there would be no real loss of 

 generality if we should choose, as the body to which the letters refer, such a quantity 

 of the gas that one of the constants a and c should be equal to unity. 



