in Rectangular Order upon a Medium. 499 



To deduce the formula for the kinetic energy we have only 

 to bear in mind that density corresponds to electrical resist- 

 a/we. Hence, by (26), if a denote the density of the cylin- 

 drical obstacle, that of the remainder of the medium being- 

 unity, the kinetic energy is altered by the obstacles in the 

 approximate ratio 



( -+i)/( O -i)+ y 



( <r+ i)/( ._i ) _ i , i > 



The effect of this is the same as if the density of the whole 

 medium were increased in the like ratio. 



The change in the potential energy depends upon the 

 " compressibility " of the obstacles. If the material com- 

 posing them resists compression m times as much as the re- 

 mainder of the medium, the volume p counts only as p/m, 

 and the whole space available may be reckoned as 1 — p +pjm 

 instead of 1. In this proportion is the potential energy of a 

 given accumulation reduced. Accordingly, if p. be the refrac- 

 tive index as altered by the obstacles, 



fi 2 = (ll)x(l-p+plm). . . . (72) 



The compressibilities of all actual gases are nearly the same, 

 so that if we suppose ourselves to be thus limited, we may 

 set ra = l, and 



, _ Q+l)/(<7-l)+ ;) . 



" ~ O + D/Or-lj-^ • • • • (73) 

 or, as it may also be written, 



— r, — t — = constant (74) 



In the case of spherical obstacles of density a- we obtain in 

 like manner (m = l), 



(2 - + l)/( q — l)+p 

 " ~ (2<r+l)/(<r-l)--V • • • ^ 

 or 



%, — .- - = constant (76) 



In the general case, where m is arbitrary, the equation ex- 

 pressing p in terms of p? is a quadratic, and there are no 

 simple formulae analogous to (74) and (76). 



It must not be forgotten that the application of these 

 formulaa is limited to moderately small values of p. If it be 



