68 EEPORT— 1891. 



medium to its original state — by pushing the sphere left to 

 right a distance a — is : 



//: 



* (El 6 .2rcos<^ — Eira;coS(^)f7a:. c7s 



— where the first integral is to be taken so as to include all 

 the tubes that make up the sphere. 



This integral =Ei(a5— ^"jxV (the volume of the sphere ; 



for / 27-cos<^fZs = V). 



Again, to replace the sphere of elasticity E.,, displaced a-\-h 

 to the left, by a sphere of elasticity Ej displaced h to the left, 

 requires an extra amount of energy. 



VEi.l-VE.. (^ + ^)' 

 ^ U" "2 



a^ ah 

 = VE,.^-VE,.— |-^, 



since E, {a+h) = E^ -| i -q } • 



Adding these, we find that the loss of energy occasioned by 

 the presence of the sphere 



= E,.^.V 



4 



=:47rr.Ex. ^'-'^' xh\ 

 2(Ei-2E,) 



For the purpose of comparison with the result obtained by 



the other method, put — for Ei, ^ ; for E.^, -^ ; and for E^S, F. 



We then obtain 



Difference of energy = i^ 1- — i-^ . Kj . F- . 



='•' 2(2 Ki + K,) 



This does not quite agree with Equation 118 on page 128 of 

 Gray's " Absolute Measurements in Electricity and Magnetism," 



as we have here an extra factor, — ^ . The Ki, however, is 



necessary, and is evidently an accidental omission in Gray's 

 form of the equation : without it the dimensions of the equa- 

 tion are wrong. I do not know why the factor h occurs in this 

 work and not in Gray's. 



Whether the — be present in the equation or not, the re- 



suits obtained by Boltzmann are unaffected ; as he uses the 

 ratio of the differential of the above expression obtained in 



