IN THE INTERIOR OF TRANSPARENT BODIES. 419 



In exactly the same manner the equations {F) may be adapted to 

 the case of spherical waves. 



§ 25. From the results in Art. (22) we have for plane waves in 



vacuum, 



1 d 2 a . d'a j • i P d*(i , d°y 



r-r = A -y-B , and similar expressions for -rr , and —r-r , 



m df dw r dv or 



1 d"a n d 2 a , .. , d'fi , d 2 y 



m~f mill era i-ir\ i lm< *"\-v nn/M>firtVin 4 ; \i* . . t . r\ * 



when the vibrations are normal, 



and — j— = B -=-= , and similar expressions for ^V^ . and ^-? 

 »» dtf du* r d/ 2 rf/ 2 



Hence it follows that transverse and normal vibrations are propa- 

 gated, in general, with different velocities, namely, \Z~A~ and */ B 

 respectively. 



§ 26. If the medium be capable of transmitting transverse vibra- 

 tions only, we must have B > 0, A = 0, or < 0, and, of course, the 

 equilibrium stable. Or by § (11), and note § (16), 



Hh d -T}>°--*- «^}-*-<. <* 



and 2{i^3}>0 (3). 



A number of laws of force might evidently be found satisfying these 

 conditions. 



Two laws of force have been a good deal insisted upon, namely, 

 the inverse square, and the inverse fourth power : these conditions 

 shew that neither of these laws can hold (assuming the theory of 



transversal vibrations) ; for if R = ± — , — ^ — - = 0, and therefore (3) is 



d( 7?T i \ 



not satisfied, the equilibrium is neutral: and again, if R= ± -j, - — -j — ■— 0, 



and therefore (1) is not satisfied, the velocity of transverse vibrations 

 is zero. Hence neither of these laws of force will answer. 



2 



- is the maximum value of a for spherical waves ; and therefore -j , the intensity of light 

 diverging from a point, which therefore varies as (distance)"'. 



