A. Vaughan — Problem of a Cooling Earth. 507 



object could be attained in two ways ; eitber by splitting, tbat is, 

 by overcoming the forces of cohesion between separate particles, or 

 by moving bodily inwards, so as to become flatter and therefore 

 smaller. The resistance to this last-named effect will arise from 

 the pressure of the interior sphere, and in the case we are consider- 

 ing, of an unequally contracting shell, the effect will be to subject 

 the interior sphere to pressure at all points, but to much greater 

 pressure under the more rapidly contracting areas than under those 

 whose contraction is less. The result seems clear, that the interior 

 sphere will become deformed ; material being transferred from under 

 the areas of great pressure to beneath those of less, with the after 

 effects I have stated in a former paper. The mechanism by which 

 the solid interior sphere would thus become deformed can be illus- 

 trated by the consideration of two layers of closely touching 

 spherical balls. If we suppose pressure to be applied over any 

 area of the upper layer, the lower layer of balls beneath that area 

 will be pushed asunder, and those which lie on the borders of the 

 area of pressure will wedge themselves between the layers of balls 

 surrounding the ai'ea, and thus produce a vertical elevation around 

 the area of pressure. 



Secondly, suppose an ideal case, that the shell contracts equally 

 at all points equally distant from the centre. It follows that the 

 pressure exerted upon the interior sphere will now be the same 

 at all points of its surface, and, consequently, no transfer of material 

 from areas of high to areas of low pressure can. take place. The 

 only relief to the forces of contraction can now be found in rupture 

 or deformation. 



Either of two effects may ensue. In the first place, the shell 

 may, in contracting, split and each separate portion then contract 

 normally, so that its contraction in any direction is linear. It is to 

 be remarked that any want of homogeneity, or, in other words, any 

 slight differences in contractibility, which may exist among the 

 materials of which the shell is made, will aid in producing this 

 effect. Secondly, the shell may stretch without splitting. This 

 is a case of great interest as being the one usually assumed in 

 problems dealing with the contraction of the earth's crust. 



This assumption amounts to supposing any shell to contract by 

 splitting, and then the separate portions to be pressed together 

 again by the pressure of the overlying mass ; much as if a very 

 heavy roller had been pushed over their surface. 



Now this second process will obviously evolve heat by the mutual 

 rubbing of particle over particle ; so that each shell must be re- 

 garded as an origin of heat, of which no account is taken in using 

 the ordinary differential equation of conduction. The general effect 

 will be to increase the time from first consolidation. 



For, let us suppose the two processes to take place perfectly 

 independently. First, let us suppose the shells to contract by 

 splitting, so that no portion is in any way deformed, but decreases 

 in volume, always remaining similar to itself. If, then, we further 

 assume the conduction of heat to be unaffected by the vacuities so 



