18 REPORT 1859. 



is unnecessary, in using this exponent, to go beyond the first place of decimals. 

 The exponents for the various media, calculated from this equation, are given in 

 Table I. The constants e n and ttn are determined from the nth power of the nor- 

 mals, in the same manner as e and a. are determined from their first power. These 

 values are also given in Table I. 



The logarithms of the normals for each exponent from 1 to 3*5 are given in 

 Table V., the normals adopted being those determined by the method explained in 

 the separate paper on that subject. 



The indices of refraction for the various media, as calculated from the general for- 



\n 



inula / i=X»-j-__ - etn, are given in Table VI. ; the observed indices in Table VII. ; 

 en 



and the differences between the two in Table VIII. 



These tables are next minutely analysed, the observations being for this purpose 

 classified. From this analysis it appears that, as respects the observations of 

 Fraunhofer and Rudberg, the agreement between the observed and calculated indices 

 is so close as to leave no doubt of the accuracy of this exponential law ; that as re- 

 gards the larger proportion of Powell's observations, the agreement is equally satis- 

 factory ; but that in some of these the discrepancies are considerable. It is proved, 

 however, by comparing the different observations together, that these discrepancies 

 can be attributed only to errors of observation. For example, it is pointed out that 

 while Fraunhofer's two sets of observations on water agree almost perfectly with 

 the law, Powell's single observation on the same medium exhibits a very consider- 

 able discrepancy, which can be attributed to nothing but experimental error. It is 

 next shown that Powell's observation on oil of cassia at temp. 14° Cent, presents a 

 discrepancy from the law scarcely exceeding that of his observation on water, so that 

 it also may be fairly attributed to experimental error. But his observations on oil 

 of cassia at temp. 10° and 25° present the greatest discrepancies of all from the law ; 

 and this difference between the results obtained for the same medium at these different 

 temperatures can be due to nothing but experimental errors, seeing it is the observa- 

 tion at the intervening temperature that is least discordant with the law. Thus, if 

 the larger discrepancies, in the case of the oil of cassia at the extreme temperatures, 

 be traceable to errors of observation, all the smaller discrepancies in other media may 

 be fairly attributed to the same cause. 



When the indices of all the media have been corrected by the exponential law, 

 then the whole become quite regular, as respects the position of the nodes of the 

 extrusions and the relations which these quantities bear to each other, with the 

 single exception of the oil of cassia; and as considerable errors of observation are 

 shown to exist in that case, it appears not improbable that this exception might be 

 removed by a more careful repetition of the observations. 



The exponential law is then contrasted with the hypothesis of M.Cauchy, — namely, 

 " that the differences between the refractive indices of the medium are to each other, 

 very nearly, as the differences between the reciprocals of the squares of the normal 

 wave-lengths. Or the refractive indices are each composed of two terms, whereof 

 one is constant for the medium and temperature, the other reciprocally proportional 

 to the squares of the normal wave-lengths." The indices calculated by Powell on 

 the basis of this law, are compared with those calculated on the basis of the expo- 

 nential law, and the differences are presented in Table IX. The result is shown to 

 be greatly in favour of the exponential law. In the case of Fraunhofer's observa- 

 tions, the rate in its favour, as compared with the law of M. Cauchy, is as 2 to 1 ; 

 in tludberg's observations as 4 to 3, in Powell's as 10 to 7, and on the aggregate 

 nearly as 3 to 2. In the particular and important case of the bisulphuret of carbon, 

 the rate exceeds 5 to 1. 



The two laws are next examined and compared, as respects their principle and 

 physical interpretation. The law of M. Cauchy merges together all the three phe- 

 nomena — the refraction, the dispersion, and the irrationality, as if they were all due 

 to one and the same cause ; and it seeks, by a general formula, to dispense with 

 observation to a certain extent, and to find the refractive indices of four of the fixed 

 lines, from those of the other three being given by observation. In the exponential 

 iaw, on the other hand, the refraction, the dispersion, and the irrationality are 



