G. F. />' < h /• — Proof of tfa Earth? 8 Rigidity. 8 ! . 



ni;i\ note that, judging from the value of "Young?s modulus," 

 cast iron and phosphorus bronze haw. a rigidity nearly the same 

 'liar of brass. The rigidity of brass is from a determination 

 by Everett In the table n g is the value of n in grams 

 weight per square centimeter; e t ie the ellipticity of a homoge- 

 neously Btrained Bphere, treated in this paper as affording a 

 rough approximation; > ,. is the ellipticity as given by Thom- 

 son^ formula for the case of incompressibility ; e c is the value 

 given by his formula for the case of compressibility if 3#= 5n, 

 or if isotropy involve but a single constant. Finally S and R, 

 which will be of use presently, are given by 



s-_^_ and R = 



ELiiPTiomEfl op Elastic, Non-Gravitating Spheroids. 



e, e r e c 8 R 



Glass 244x10* 235 x 10*t 248x10*7 257 x 10*r 0591 0*606 



Brass :;;:ixlir 153x10*7 161 x 10*r 167xl0*r 0485 0-498 



I 780x10* 72xIO*t 77 x 10 4 r 80 x 10V 0-311 0-322 



Comparing these ellipticities with that for a fluid sphere, 

 = 162x10V, it appears that in a globe of the size of the earth, 

 the mutual attraction of parts is ranch more powerful than the 

 elasticity of figure if the rigidity is that of glass ; that these 

 t\v<> resistances to deformation are substantially equal if the 

 rigidity is that of brass : and that gravitation is less than 

 half as powerful as elasticity of figure if the rigidity is that of 

 BteeL* 



ibination of resistances. — The ellipticity of a deformed 

 sphere varies directly as the applied force and inversely as the 

 ~ance. Thus if AY is the resistance one may write 



1 W 



In the cases of an elastic incompressible sphere and a fluid 

 Bphere this equation take.- the forms 



1 19>i 1 r ,1 2g 1 g 

 5a*w ' t ~ ~ t ' e„~" 5a ' r r 



where the Old English letters are introduced for brevity. In 

 the case of a sphere in which gravitation and elasticity co-ope- 

 rate. AY = g + x and the ellipticity is expressed by 



* In the first sent' Nat Phil., § 340, to m€ certain that 8ir Wil- 



liam Thomson must have meant to write da— instead of Bteel, for he has just 

 shown that the attraction of the moon deforms a fluid Bphere much more than it 

 does a steel sphere in which there is nu mutual attraction of parts. 



