of Chromatic Dispersion. 267 
Among the fourteen media in which the discrepancies are 
greater, there is found water, as observed by. Powell, in which 
the total errors amount to no less than 0:001916; whereas in 
Fraunhofer’s two sets of observations on this medium, their 
amounts are 0:000154 and 0:000205,—Powell’s discrepancies 
exceeding the least of Fraunhofer’s by 0°001762, an excess which 
can be due to nothing but a difference in the degree of accuracy 
with which the observations were made. Thus the total discre- 
pancies of 0001916 in Powell’s observations on water are clearly 
traceable to experimental inaccuracy. But the total discrepan- 
cies in the case of Powell’s observations on oil of cassia, temp. 
14°, very little exceed this amount, being 0-001984; so that 
these may also be fairly attributed to the same cause. Now the 
reasoning applicable in the case of the oil of anise applies equally 
to the observations on oil of cassia at temp. 10° and temp. 
22°-5. The discrepancies in these two cases amount respectively 
~ to 0:003750 and 0:003529, or not far from double of what they 
are at the intermediate temperature 14°. This difference can be 
attributed to nothing but the inferior accuracy with which the 
observations at temp. 10° and temp. 22°°5 were made ; and had 
only the same amount of care been bestowed on these as on those 
made at temp. 14°, the gross amount of discrepancies would not 
have exceeded those presented in the latter case, which have 
already been shown to be due to experimental error. Thus the 
extreme amount of the discrepancies in the case of oil of cassia, 
temp. 10°, may be logically traced to defective observation ; and 
these discrepancies being the greatest in the Table, it may hence 
be quite fairly inferred that all those of lower amount ought to 
be attributed to the same cause. 
The indices, calculated by the exponential law from the four- 
teen observations of the first order, may be regarded as being 
quite as correct as they can be possibly obtained. Those caleu- 
lated from the twenty-one observations of the second order may 
be deemed very nearly correct; while those calculated from the 
five observations of the third order may be viewed as fair ap- 
proximations to the truth. It is, however, too much to expect 
of the exponential law that it should yield accurate indices from 
the eleven observations of an order inferior to the third. No 
mathematical law whatever can bring forth accurate results from 
incorrect observations where the errors exceed a certain limit ; 
the utmost that can be expected in such a case is, that the law 
should indicate the probable position and amount of the errors of 
observation, and exhibit the necessity for more careful repetition. 
It is in this light, then, that the calculated indices of these eleven 
cases ought to be regarded. 
This point must be kept in view in examining the question, 
