332 The Rev. Professor O'Brien on the 



which, equating possible and impossible parts, gives 



A{k^-h^) = 7i^ + ^n, (5.) 



T 

 2A^A = — «; (6.) 



and hence we find 



2 AF = « { ^(,. + ff + (Tf + {„ ^ ^)}, (7.) 



We reject the negative sign of the radical in these expres- 

 sions because it would make k impossible; also, since A, k, T 

 and V are essentially positive, equation (6.) shows that // must 

 be positive also. It appears therefore that the equation (4.) 

 admits only of one solution, namely ae~^^, where A is a posi- 

 tive quantity given by equation (8.) and a an arbitrary con- 

 stant. The results we have thus obtained may be found im- 

 mediately by putting ^ + ,3^/^ = ae(«^-*^)V-i-Az j^ the 

 equation (S.). 



11. We have assumed that u (the radius of the circle of vi- 

 bration) suffers no sensible change when z increases by a wave's 

 length (A suppose); hence the ratio ae~^^: ae~^(^+^) must 

 be very nearly unity, and therefore h A must be a very small 



quantity : now X = -7-, therefore -r- must be a very small ratio, 



and, it is manifest from the equations (7.) and (8.), that this 



T . 

 will be the case when — is very small compared with n or with 



N 

 n -i . Now w, as is well known, is an extremely large 



number, therefore all that is necessary to the correctness of 

 the above integration is this, that the tangential resistance (T) 

 be not very large compared with the velocity of vibration (t;). 



12. Since h is essentially positive, it follows that m continu- 

 ally diminishes as the wave advances, and it appears from the 

 equation u = ae~^'^ that the rate of decrease of 7/ is hu; h is 

 therefore the proportional rate of decrease of the amplitude of 

 vibration. 



13. We have thus obtained the laws of propagation of cir- 

 cularly polarized light in uncrystallized substances, which may 

 be stated thus: 



2 TT 



The time of vibration being — , the velocity of propagation 



