248 Mr. C. Chree on some Applications of 



For a given value of 77 the value of S is independent of E. 

 It diminishes continually as 77 increases from zero. Since 

 the value of s depends on E, I have given the value of Es, or 

 the longitudinal stress which would produce in a bar of the 

 material a strain equal s. The value of E " has a maximum 

 of about 1520 tons weight per square inch, for 77=1— \/ 1/2 

 or * 2 93 nearly. 



For 77 = *5 the values of s and u a are zero supposing E 

 finite, but for other values of 77 one can obtain numerical 

 measures of these quantities only by assigning numerical 

 values to E. Now if the earth were an elastic solid truly 

 spherical but for its rotation, the value of E answering to a 

 given value of 77 would be determined from the eccentricity 

 of the surface. But the action of the gravitational forces, as 

 will be seen more clearly presently, largely reduces the eccen- 

 tricity which rotation would produce in a sphere of given 

 material. Thus the eccentricity varying inversely as E, the 

 value of E answering to a given eccentricity is necessarily 

 considerably smaller when gravitational forces act along with 

 the " centrifugal " than when the latter act alone. Since the 

 surface-strata are very variable and of much smaller mean 

 density than the earth as a whole, any calculation of the 

 reduction of our estimates of E, when gravitational forces are 

 allowed for, which treats the earth as of uniform density can- 

 not lay claim to great accuracy. For this reason, and also 

 because I am specially desirous not to overstate the case 

 against the application of the mathematical theory, I have in 

 calculating the values of s and u a in Table IV. ascribed to E 

 the values it would possess in the total absence of gravita- 

 tional forces, viz. the values 1020 x 10 6 for 77 = and 1220 x 10 6 

 for 77 = *25 in the same units as before. The numerical values 

 ascribed to s an d u a in the table are thus essentially minima, 

 which w r ould in reality have to be increased probably to a 

 considerable extent. 



It will be seen from the formulas and from Table IV. that 

 when 77 is zero or is small, the application of the mathematical 

 theory would be fully justified on the greatest strain theoiy, 

 w r hile utterly condemned on the stress-difference theory. 

 The principle (C) is in this case entirely in agreement with 

 the stress-difference theory, and the application of the mathe- 

 matical theory can in fact be supported only by those who 

 reject this principle, and consider it possible for the stress- 

 strain relation to remain linear though a solid sphere is 

 reduced to one fourth or less of its original volume. 



Noticing from (1) and (2) that E"/S = 2?7, we see that for 



