of Chromatic Dispersion. 271 
tial law, the exponent for this medium being 2°5, are 
*B+.0-000143, *C—0-000252, “D+0-000047, “E—0-000110, “F—0-000295, 
#G—0:000439, 4H-+0:000390, S+0-001675. 
So that the sum total of these latter is less than the amount of the 
single error in D in the former case, and less than a fifth of the 
total errors arising under the law of M. Cauchy ; consequently, in 
this important medium, the ratio in favour of the exponential law, 
as compared with that of M. Cauchy, is more than 5to 1. The 
difference is still more striking if the individual discrepancies be 
compared,—the highest arising under the exponential law being 
only a tenth of that arising under Cauchy’s law, the latter dis- 
crepancy, moreover, being far too large to be attributed to errors 
of observation ; while those arising under the exponential law 
are all of such moderate magnitude, that there can be no hesita- 
tion in ascribing them to that cause. 
From an inspection of Table IX. it will be seen that, as respects 
Fraunhofer’s observations, the agreement between the calculated 
and observed indices is as 2 to 1 in favour of the exponential 
law*. In Rudberg’s observations the ratio is as 4 to 3, and in 
Powell’s as 10 to 7, while from the three sets combined it is as 
6 to 4. But the best criterion of judgment is furnished by those 
media which have a high dispersive and extrusive power, and in 
which the law of M.Cauchy entirely fails, presenting discrepancies 
far too great to be attributed to experimental error. Such are 
those in the case of the bisulphuret of carbon above noted ; such 
are also the large discrepancies in the case of the oil of cassia, 
ranging between 0:0017 and 0:0029, while the largest indivi- 
dual discrepancy arising under the exponential lawis under 0-001. 
In some few instances it will be observed that the result appears 
to be in favour of the law of M. Cauchy, but these anomalies are 
all clearly traceable to experimental error. Looking at the 
results as a whole, there can be no doubt that the decided supe- 
riority rests with the exponential law, as being the true law of the 
indices. 
The great defect in the hypothesis of M. Cauchy is its failure 
to accommodate itself to the phenomenon of irrationality and the 
attendant extrusion of the fixed lines. Its apparent agreement 
with observation in a considerable number of cases, arises simply 
from the circumstance that, with the squares of the normals, the 
extrusions are in those cases so small that they may be elimi- 
nated without greatly affecting the indices ; and it is only when 
* It must be kept in view, in examining this Table, that the normals on 
which Powell’s calculations are based differ slightly from those specified in 
this paper; but this circumstance does not materially affect the general 
results. 
