i;)6 UXDULATORY THEORY OF LIGHT. 



tion of tlio question in its most general form. If we put A for the wave length 

 and T for the time of vibration, and assume — 



2- 2;r 



X T 



tlie n>latiou between tliesc quantities is iinnid by liim to be expressed by the 

 formula — 



in •wliicli the constants fii, a2. (I3, &c., arc the same for all rays in the same 

 medium, but ditfer for different media. 



Now the velocity of every uniform movement is expressed by dividing the 

 spaci' p;!ssed over in any given time by the time of passing, and assuming a 

 Avave length for the space, and V for the velocity — 



2- 

 V ^= - = 5- ^= T • Ilencc, kzzzY' «in<-l /c" r:^ Vr •>, 



and, by substitution, 



Y^=ai + a-Jc^ + ojc^ + &:c. 



The velocity in the same medium is therefore a function of the length of the 

 Travc. But if, in any medium, the coefficients of the terms beyond the iirst are 

 insensible, the equation will become \'^::izai, or the velocity will be constant for 

 all the rays. This he supposes to be true in a vacuum and in media which, 

 like the air, do not sensibly disperse the light. " 



13y reverting the first series it becomes — 



wliich is convergent like the first, and in which all the terms after the first three 

 may be neglected. 



Now, as ^^^^^r> if the velocity in vacuo be put equal to unity, we shall have, 



by substitution and reduction, 



TTi^^/^-^^Ai + Aos-' + A-.r. *■ 



V 



in which ,a is put for the index of refraction. This index is therefore a function 

 of the time of vibration in vacuo, which timi^ is necessarily unaltered by refrac- 

 tion ; since the period in the second medium is determined by the period of the 

 impulses, which impulses are the vibrations in the first. But the time of vibra- 

 tion in vacuo is inversely as the length of the wave. Ilenoe, if we determine 

 by observation three wave lengths in vacuo, and three corres})onding indexes of 

 refraction, we shall have the data for determining the three unknown constants, 

 A], Aa, and A3. For all other wave lengths the indexes may then be computed. 



Tlies(^ theoretic conclusions can only be thoroughly tested by comparing the 

 values deduced from the formula with the results of observation upon media of 

 high dispersivt! powers. A very elaborate series of such comparisons was made 

 by Prof. Baden Powell, which exhibits a general agreement between the com- 

 puted and observed values quite within the limits of the probable errors of ob- 

 servation. We are therefore justified in saying that the different velocities of 

 propagation of luminous undulations of difierent lengths no longer constitute a 

 serious objection to the undulatory theory. 



It will now be easily seen that so long as the movements of the vibrating 



