INTRODUCTION, 177 



The second class of equations, in addition to (1), (2), includes the equa- 

 tions of the particular path of compression in question, together with the 

 expression for the work done in compressing unit-mass: 





e= / -^,dp (12) 



'Pi 

 which gives through integration by parts the relations 



ep-^p=ps d(ep)=sdp (13) 



the useful auxiliary variable s being defined by 



s= r^dp (14) 





where the elastic bulk-modulus H is 



i/=.f (.5) 



The limits of the integrals correspond to the supposition that the mat- 

 ter is compressed from surface density and zero pressure. With e so deter- 

 mined, and expressed in terms of r by means of equations of the first class, 

 the total energy of compression is 



/edm=4:7z I epr^dr = — ^ / s 



E= I edm=^7z I epr'dr = -^ / sp'^dr (1^) 



the last form being obtained by integration by parts and equation (13). It 

 may be noticed that the quantities ku and s are of the same physical dimen- 

 sions, but with distinct theoretical setting; in case, however, the two paths 

 on the thermodynamic surface are identical, they are equal at every point 

 in the body. In this special case the value of E may be given as 





E = *^ I r'(F,-y)fdr 



An essential feature of the present hypothesis is the necessity of suppos- 

 ing that most of the energy of impact is wasted by immediate radiation, so 

 that the compressional energy whose total is E plays the main part in the 

 succeeding phenomena. It is therefore important for purposes of compar- 

 ison to determine what ratio the quantity E bears to the total energy 

 transformed. 



An idea on this point may be obtained by considering the case of a planet 

 formed by condensation of a primitive homogeneous sphere of density p^. 

 Suppose that a particle at distance r from the center in the completed planet 

 lay momentarily at distance r' when deposited at the temporary surface, 

 and in the homogeneous sphere would lie at distance r,^, these being subject 

 to the inequalities r^>r'>r. It may be considered that the particle fell 



