146 Rev. 0. Fisher on Elevations attributable to 



done in my former work, that the matter in each layer retains 

 its horizontal extension during the settlement into its present 

 position. On this supposition the corrugations will clearly be 

 influenced by the sphericity of the surface. But if we make 

 use of Sir W. Thomson's expression* for the temperature at 

 any depth, we must recollect that he neglects the sphericity. 

 Still it seems probable that his law of cooling for an infinite 

 plain will be sufficiently applicable to the globe to make the fol- 

 lowing of some value. For it is evident that the temperature- 

 curve for the sphere will be of a similar character, though not 

 exactly of a similar form ; the more rapid escape of heat to- 

 wards the convex surface causing the ordinates to decrease 

 somewhat more rapidly as the free surface of the sphere is 

 approached. 



Let a layer of the globe at a distance z descend, by cooling 

 of the matter beneath it, to the distance z from the centre C. 

 Then our assumption, that this layer retains its horizontal 

 extension, necessitates that we suppose the voluminal con- 

 traction to take place wholly in the vertical dimension. 



Let E be the coefficient of voluminal contraction. If, then, 

 the layer in question has fallen through 6° since it solidified, 

 we must have 



dz=(l-R0)dz; 



or dz'=(l + ~Ej0)dz, approximately. 



The volume of this layer on first solidifying was 



47T.2 /2 ^'. 



And, after cooling, the thickness of this layer has contracted 

 to dz, but has retained its horizontal extension. Its volume 

 therefore becomes 



47Tz"*dz. • 



Also, the proper volume of the spherical layer of the same 

 thickness at this depth is 



47r z 2 dz. 



And the difference between these volumes will be the contri- 

 bution to the surface-corrugations from this particular layer. 

 Call the volume of the whole corrugations iirr^h ; we then 

 shall have 



4:irr 2 -^dz = iirz'Hz— kirzHz. 

 dz 



But since every layer beneath the one in question has cooled 



* Trans. Koy. Soc. Edin. vol. xxiii. pt. 1, p. 157. Also Nat. Phil. 

 Appendix D j and Phil. Mag. 4th series, vol. xxv. p. 1 (1863). 



