264 Mr. M. Ponton on certain Laws 
the two sets agree very closely, this agreement does not extend 
beyond the fourth place of decimals; so that in no case can the 
fifth and sixth places of decimals, as given by observation, be 
depended on, while it is needful to carry the indices to the sixth 
place of decimals in order to their fulfilling with exactness the 
exponential law. But if the observed indices be correct down to 
the fourth place of decimals, the exponential law, in combination 
with the laws governing the extrusions, may be relied on for the 
fifth and sixth places. 
It is proposed, then, to consider all those observations in 
which, when tested by these laws, no individual error amounts 
to 0:0001, as of the first order, greater accuracy of observation 
being unattainable ; those in which no individual error amounts 
to 0:0002, as of the second order; those in which no individual 
error amounts to 0:0003, as of the third order, and so on. 
As regards Fraunhofer’s observations, it will be found that, of 
the whole twelve, there are of the first order seven; namely, 
water (two sets), solution of potash, oil of turpentine, and three 
specimens of crown-glass; while there are of the second order 
five, all of them on flint-glass. 
With respect to Rudberg’s ten observations on doubly-refract- 
ing media, there are of the first order seven, namely, topaz 
2nd axis, quartz ex. ray; Arragonite Ist axis, quartz O. ray; 
topaz 3rd axis, cale-spar O. ray; topaz 1st axis: and of the 
second order three, namely, Arragonite 83rd axis, calc-spar ex. 
ray, and Arragonite 2nd axis. 
With Powell’s observations the results are not so satisfactory ; 
but that the discrepancies which they present are due, not to any 
peculiarity in the media, nor to any defect in the exponential law, 
but simply to the inaccuracy of the observations themselves, may 
be clearly shown. It fortunately happens that we have a set of 
observations by Powell on water, on which we have two sets by 
Fraunhofer. While the two latter agree very closely with each 
other, and quite as closely with the exponential law, both being 
of the first order, those of Powell are so inaccurate that they can 
be classed only as of the eighth order. Now this difference can 
be attributed to no other cause whatever than to the inferior 
accuracy of Powell’s observations; for it appears highly impro- 
bable that it should be due to the difference of temperature at 
which the observations were made. In the case of solution of 
potash, on which we have also observations both by Fraunhofer 
and Powell, and at temperatures more widely apart, the difference 
in quality between the two sets is much less marked; for while 
those of Fraunhofer are of the first order, those of Powell are 
of the second. 
But if in so simple a case as that of water the observations of 
