THERMODYNAMIC THKORY, 221 



by substitution of the value of -^ from the equation of conduction, while 

 the total rate of work of gravity is 



-^ = -A. f\nKur (129) 



=-"/"^ 



J 



which by equation (3), through integration by parts, takes the alternative 

 forms ^y.^ ^f^ 



d0 



dt 



= -4./';al(..,^)*=4./V|i^)* (130) 



This gives simply the total rate at which energy is transformed from the 

 potential form, while the manner of its localization remains undetermined. 

 The last integral, which contains only thermodynamic quantities, suggests 



. , , ,3^ d(pa) , , 

 that the rate of transformation per unit volume might be X-r- ^ , but 



any such special interpretation is purely arbitrary as long as the thermo- 

 dynamic substance is so incompletely defined. 



The rates of specific linear expansion horizontally and vertically are 



The sum of the latter and twice the former gives by (128) the rate of spe- 

 eific volume expansion a— as it should. The moment of inertia is 



1=^^ TpT'dr (132) 



and since d(pr^dr) =0 its variation is 



pr^ dr dr 



dl 



dt 



which gives 



^'P f\^r f\l{..^yr (133) 



t/ t/ 



Any supposition as to how a varies with the depth would appear to be 

 wholly gratuitous, but it may be worth while to follow out a simple one 

 suggested by the form of the equations, namely, that a is constant. Under 

 this condition the radial motion is 



dr ,dd 

 dt dr 



which is constantly negative at all depths, under all of the hypotheses en- 

 tertained above in the computation of the temperature-curves, so that the 



