176 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



In the first class come the equations: 



dp 

 dr 



where 



= —gp with 7)1=0 (3) 



km ... 



9=-^ (4) 



and 



=47z I prHr (^) 



J a 



m 



Jo 



which express the condition of hydrostatic equihbrium and yield the im- 

 portant relation: 



d /r"^ dp 



dr\p 

 subject to the conditions 



P.=0 , f ) =0 



{ith'^^"'-"^ («) 



To these must be added the expression for the potential energy ex- 

 hausted during contraction from infinity: 



0=^h I Vdm=27ik I pVr' 



'dr (7) 



where 



V=— I pr^dr-^in I fyrdr (8) 



/r nr^ 



pr'dr-\-47: I prdr 

 Jr 



the latter being equivalent to 



.^ = -, (9) 



with 



•f 



fx 



yj=4;r / prdr or ^1 = 



Let 



u = Y-Y^ (10) 



then equations (9), (3) give 



^(,= ^)+4.^.=0 (11) 



where p may be considered expressed in terms of u or r, since u is necessarily 

 and p most probably a monotonic function of r in the concrete case. 



The preceding equations then suffice to determine in terms of r all vari- 

 ables involved, if supplemented by a single hypothetical equation, such as 

 an expression for p in terms of r or a relat-.on between p and p. 



