CRITICAL AND SUPPLEMENTARY. 211 



within the earth. It is here dwelt on briefly for the sake of the comparison 

 it affords with still another now to be considered, which also belongs to the 

 class for which , 



^-V (104) 



of which (20) and (96) are examples. 



It was shown in Part I that, with the particular law of compressibility 

 there postulated, the progressive condensation under the increasing load of 

 the material gathered at the surface would be accompanied by a deforma- 

 tion of the elements of the mass whose final amount is indicated by the 

 distortion-factor tabulated. Attention was called to the consequent uncer- 

 tainty introduced into the determination of the work of compression if it 

 be supposed that the substance offers appreciable resistance, either elastic 

 or viscous, to shearing stresses, so that the working pressure is not purely 

 hydrostatic. It is conceivable, however, that this deformation might be 

 widely different in character and amount under another acceptable pres- 

 sure-density law, so that the acceptance of the special formulas (20) and 

 (25) would lead to no just estimate of the essential obscurity in the theory 

 from this source. 



As a guide to conjecture on this point it will be of interest to determine 

 whether there could be a law of compressibility assumed of such nature 

 that the condensation of the mass would lead to no such deformation, but 

 rather that the compression would be at all points purely cubical. 



If X be an auxiliary variable, determining the location of a given par- 

 ticle at some chosen epoch, for instance as before the ratio of the ultimate 

 distance from the center to the total radius, then the distance r from the 

 center at any epoch may be considered as a function of a and x: 



r=<Pia,x) (105) 



Let x', x" be the values of x corresponding to two chosen particles; then 

 the mass of the spherical shell whose bounding surfaces pass through those 

 particles will be „ 



m(x',x")=4;r / pr^^dx (106) 



;',x")=4;r / 

 Jx' 



As the condensation progresses the spherical surfaces will shrink, but the 

 mass between them must remain constant. This means that the integral 

 (106) must be independent of a, whatever the values of x', x"; this gives 

 the condition 



Br { f dp dp dr\ dr ] dV . 



which reduces to 



in which is put 



^(a,0=|^ (108) 



14 



