TRANSACTIONS OF THE SECTIONS. 17 



from the operation of the dispersive power alone. The actual loss of length, being 

 represented by ect, is the same for all waves ; consequently it tells more on those 

 waves which are primarily shorter. Hence the numbers representing the loss of 

 length sustained by each wave in proportion to its primary length, from the ope- 

 ration of the dispersive power alone, are in inverse proportion to the primary wave- 

 lengths. These rateable losses of wave-length, multiplied by the second series of 

 refractive indices fi 2 B, ft 2 C, &c, exhibit the proportion of each index due to the 

 dispersive power alone. These points are exemplified by the case of the bisulphuret 

 of carbon. 



In different media, the loss of length, sustained by any one wave through the 

 operation of the dispersive power alone, is proportional to the constant ct, which may 

 accordingly be regarded as its measure. 



The effects of change of temperature are illustrated by the two cases of oil of 

 anise, and oil of cassia, for which alone sufficient experimental data exist ; and it is 

 shown to be probable, that, in the same medium, the values of e are in the inverse 

 orders of the temperatures, and their differences proportional to the differences of 

 temperature ; also that, in different media, in which the value of e is nearly the 

 same, the fall in that value is proportional to the rise of temperature. 



The constants e and a., being influenced only by the mutual relations of the extru- 

 sions, and not by their absolute values, are consistent with an indefinite number of 

 sets of indices of refraction; so that the indices may always be altered in a certain 

 manner, without altering the constants. It is then shown, that the conservation of 

 the total vis viva of the normal wave-lengths depends on the constants e and u, and 

 a third constant r\, thus: 



f B C D E . F 



\(B-eJ)+ij (C-ec)+i; (D-etO+ij (E-ee) + v (F-f/)+>, 



, G H \_ g 



(G-*$0±>? (H-e/0±v/ * 



To find n, call the sum of the series „ r + ^ +&c. = 2, and call =: = €«, then n 



is the difference between ea and eti. If as > a, then rj is-f ; if a. > a, then rj is — , 

 and in either case is constant for the medium and temperature. It is always pos- 

 sible to find a positive value of X which shall render rj = 0. Calling this limiting 



value X, then is X=« 4(B + C + Q+H)-3(D+K+P 



S 

 The logarithm of this multiple of cc is 2-4216417. 



The effects of raising the normal wave-lengths to different powers are nextexamined. 

 It is shown that, in every case, there is a certain power at which the extrusions are 

 reduced to a minimum, and that these lowest values are so small that they may be 

 referred to errors of observation. There is thus always a certain exponent, which 

 may be applied to the normals, which will extinguish the extrusions, so rendering the 

 relation between the wave-lengrh\ and its refractive index p capable of being expressed 



X" 



by this general formula: /*=X»-f- — — cc n . This it is proposed to call " the exponen- 

 ts 



tial law of dispersion." 



The value of the exponent n depends on the relation which the irrationality bears 



to the dispersive power, or to the length of the spectrum. Expressing this relation 

 o v n 



aT = P' ^ e f°ll° w 'ng equation is universally applicable : • ■ -= constant. 



By analysing the observations, the value of this constant is found to be nearly 

 - 0O92593, and its reciprocal 10 - 8. These values are accordingly assumed, but sub- 

 ject to future correction. Hence 11*8 is the highest limit of the value of n, being 

 that which the medium would have if 2X = a. The lowest value of n, = 1, subsists 

 when * = 0-0092593 and 2X = 0. 



As the value of p may be obtained, with tolerable accuracy, from any set of 

 observations which are approximately correct, the value of n may be found from the 

 equation \0'8p+l=n with sufficient accuracy for calculating the indices; and it 

 1859. 2 



