Physics and Mathematics to Geology. 



247 



seemingly requires that strain should be defined as the ratio 

 of the increase of length to the final length, which is not in 

 accordance with the usual interpretation of Hooke's law 

 unless the square of the strain be negligible. Supposing the 

 internal equations to refer to the final deformed condition, 

 the surface equations will undoubtedly also refer to this con- 

 dition. Thus, so far as the terms independent of the eccen- 

 tricity are concerned, we may suppose the mathematical 

 theory applied to a sphere whose density p is uniform through- 

 out, and whose radius a equals the earth's mean radius. 



In this case the maximum stress-difference and the al- 

 gebraically greatest strain are both found at the surface. 

 Let us denote these by iS and " respectively ; and let s denote 

 the greatest compression, which occurs at the centre, and u a 

 the radial displacement at the surface. Employing E and rj 

 as before, and denoting by g the acceleration due to the 

 sphere's attraction at its surface, I find"* 



s 1 1-2*7 



b = 5 9P a 



s = — 



1-?' 

 2 gpa 77(1 — 2t?) 

 5 E l- v ' ' ' ' 



S_gpa(l-2 v )(l- v IS) 

 10 



E 



_ 1 gpa" 

 (a ~ 5 E 



(1-2*/) 



V 



(1) 

 (2) 

 (3) 

 (4) 



Assuming for a moment these results to hold for a sphere in 

 which g = gravity! at the earth's surface, p = 5'5 times the 

 density of water, and a = 3950 miles, the following are the 

 approximate numerical values answering to the values 0, *25, 

 and '5 of rj : — 



Table IV. 



n= 







•25 



•5 



S, in tons weight per square inch 



4440 

 



1-03 



2700 



2960 



1480 



•53 

 1130 













 



Longitudinal stress E s, in tons weight 1 

 per square inch, which would produce L 

 a strain " J 



— s (see below) 



— Ua, in miles (see below) 





* See (a) formulae (17), (18a), and (19a), p. 281. 



t The calculations treat the attraction on a cubic centimetre of water 

 at the surface as equal to the weight of one gramme. In reality of course 

 " gravity " includes the " centrifugal " force. 



