244 Mr. C. Chree on some Applications of 



true, for all values of rj except '5, a slight increase in the 

 volume *, and consequent diminution in the mean density 

 accompanying the rotation, but for our present purpose this 

 may be neglected. 



The mathematical solution on which Table I. is based treats 

 the spherical surface of radius a as that over which the con- 

 ditions for a free surface are satisfied. Now some uncertainty 

 may exist, depending on the physical interpretation put upon 

 the mathematical equations, whether these surface conditions 

 should be applied over what is the surface before the dis- 

 placement — in this case the surface of the true sphere which 

 it is assumed the earth would form if the rotation disappeared, 

 — or over what is the surface during the rotation. This un- 

 certainty might constitute a very serious difficulty if the de- 

 formations were supposed to be large — a contingency which 

 may arise when the limitation (C) in the magnitude of the 

 strains is neglected ; but in such problems as the present 

 where the strains are, as we shall see presently, of the same 

 order of magnitude as occur in ordinary engineering struc- 

 tures, it is of no material consequence. In the present case 

 complete assurance on this point may be derived from figures 

 1 and 2, plate ii. of (c), which show the changes induced by 

 rotation in the equatorial and polar semi-axes of spheroids of 

 various shapes. 



For given values of d, a, &>, and p, Table I. shows that E 

 and 7) increase together. Giving co the value it has for the 

 earth, and assuming p = 5*5, a = 3950, d= 13*25, I find for 

 the values of E, measured in grammes weight per square 

 centim., answering respectively to the values 0, *25, and *5 

 of rj, the approximate numbers 



1020 xlO 6 , 1220 xlO 6 and 1410xl0 6 . 



It is obvious from Table I. that to equal increments in 77 

 there correspond nearly equal increments in E ; thus the 

 numbers given above will enable a sufficiently close approxi- 

 mation to the value of E for any other value of y to be 

 immediately written down. 



For the sake of comparison with the values found for E in 

 some of the commoner materials under ordinary conditions I 

 append the following data, taken from Sir W. Thomson's 

 article on ' Elasticity \ in the Encyclopaedia Britannica. 

 The units are the same as above. 



* See (b) Table II,, and compare Tables V. and VI. of (c). 



