+e 
PHYSICS AND MATHEMATICS TO GEOLOGY. _ 139 
applied to a sphere whose density p is uniform throughout, and whose 
radius @ equals the earth’s mean radius. 
In this case the maximum stress-difference and the algebraically 
greatest strain are both found at the surface. Let us denote these by 
S and § respectively; and let s) denote the greatest compression, which 
occurs at the center, and w, the radial displacement at the surface, 
Employing E and 7 as before, and denoting by g the acceleration due 
to the sphere’s attraction at its surface, I find* 
2 ge! | 1—27 
S = 5 9p ap ; 5 = = - - (1) 
bl A cel septa een ue) 
= 5°) t=’ 
3 gpa (1—27) (1—7/3) ; : 
LEST oan te (ee a (3) 
1 gpa? é 
U,= —F ur. (eS apres er ant Re unis. UV SAE et a (4) 
Assuming for a moment these results to hold for a sphere in which 
g=gravityt at the earth’s surface, op =5:5 times the density of water, 
and @ =3950 miles, the following are the approximate numerical values 
answering to the values 0, -25, and 0°5 of 7: 
TABLE IY. 
S, in tons weight per square inch ....-..-.----- | 4440 | 2960 0 
Longitudinal stress E 3, in tons weight per} | 0 1480 0 
square inch, which would produce a strain &. 
Gn (ROE DELO) Se seis ale oe ec ctemienelewsidemia saarteme A OSee ee O:o3 0 
—ta, in miles (See DelOW) =... ---ssc-2-.c05s055-55 2700 1130 0 
Fora given value of 7 the value of S is independent ef E. It dimin- 
ishes continually as 7 increases from zero. Since the value of s depends 
on KE, 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 
Es has a maximum of about 1,520 tons weight per square inch, for 7= 
1— v1/2 or 0.293 nearly. 
For 7=0-5 the values s) and uw, are zero supposing E finite, but for 
other values of 7 one can obtain numerical measures of these quantities 
only by assigning numerical values to EK. Now, if the earth were an 
elastic solid truly spherical but for its rotation, the value of E answer- 
ing to a given value of 7 would be determined from the eccentricity of 
the surface. But the action of the gravitational forces, as will be seen 
* See (a) formule (17), (18a), and (194), p. 281. pie: . 
+ The caleulations treat the attraction on a cubic centimeter of water at the sur- 
face as equal to the weight of 1 gram. In reality of course ‘‘ gravity” includes the 
“centrifugal” force. 
