308 Prof. Powell’s further Observations on M. Cauchy’s 
mh AX ae 
v 
f(r) +o08° BS (1), 4° — ye hfe 
‘ J 
Mm 
and — 
2g 
(though H? is not necessarily positive), which, since 
Qa 6 
Ue a Ae 
will give the relation 
is sin 6 y? 
Gye id rad i, (8.) 
the same as that formerly deduced. 
Such is the outline of the simplification proposed: I have 
only to regret that these and the other researches connected 
with the same subject have not been brought before the public 
by the author himself in the more complete form in which he 
could have clothed them. But, as they are, I trustneither he 
nor the mathematical reader will think I have done wrong in 
adopting this mode of availing myself of his permission to 
make use of them. On the same ground I will add another 
brief investigation from the same source connected with the 
fundamental formula of dispersion. 
I have before observed that for low dispersive substances, 
at least, the simple approximate formula appears quite suffi- 
cient. As it may, therefore, be useful for a very large num- 
ber of cases, it will not be unimportant to dwell upon it, and 
to state a very simple practical rule resulting from it, which 
completely removes the difficulties of the computation as con- 
ducted by the methods I formerly adopted. 
In the first place, from the nature of the formula the fol- 
lowing considerations will be readily evident. Taking any two 
rays whose indices are » p,, and length of waves A A,,_ let us 
write the arc 
meCosd _ 
* = 6. 
Then we have sigicos ys = gm 
A A 
7 
and by the approximate formula 
