CRITICAL AND SUPPLEMENTARY. 213 



This could be obtained from the general equation (93) by substitution of 

 (112) and change of variables from r, a to r, ^. It shows that, with C an 

 arbitrary constant, the relation must have the form 



giving for determination of the function <p 



f" -^<p' =C<p<p''^ (116) 



on account of which (115) gives for the equation characteristic of the sub- 

 stance 



^ = hpi A=-^ (117) 



SO that C must be negative. This is, then, the condition which it is neces- 

 sary that the substance satisfy in order that the condensation under increas- 

 ing mass may not be accompanied by distortion of the mass-elements. 



Conversely, if the substance satisfy this condition, the compression will 

 take place without deformation. For, with the substitution of (117), equa- 

 tion (6) takes the form 



The solution which is finite at the center has the form 



R=R,Q(u) Ro'=Po u=R,fir (119) 



where .G is a definite function, which for sufficiently small values of u can 

 be expanded as a power series with numerical coefficients of alternate signs 



If the radius of convergence be too limited, the function may be considered 

 as determined for any value of u by analytic continuation, since equation 

 (118) has no singularities to prevent. The density is then 



p=p^oj{u) 0}{u)=[Q{u)Y (120) 



so that 0) also is expansible for small values of u as an alternating series 

 with numerical coefficients 



Then 



• • • • 



whence 



r 



Jo 



dr 

 m=4;rO(, / r-(o{u)-^du 



m^^e{u) (121) 



where 6 is the definite function 



e{u)= I uMu)du (122) 



nU 



