34 
MU. POWER ON THE ABSORPTION OF THE SOLAR RAYS, ETC. 
and in order that a usay be greater for the smaller wave lengths, as experience shows 
it is, A 2 must he positive. 
If we wish to introduce in the place of X the note-number v , or the number of vibra- 
tions performed in a unit of time, since v=~, we have 
a 2 = A 0 + ^-aV+^-aV+ &c. ; 
and since a 2 , a 4 occur in the denominators, it will be sufficient to write for them the 
first approximate values A 0 , Ay, &c., thus we get 
o*=A 0 +^«V+ji«V+&c. 
Cauchy found by comparing theory with experiment that the two foremost terms 
of his series were sufficient to account for the chromatic dispersion. We shall have, 
therefore, a sufficiently accurate value of a by extracting the square root of this series 
to the exclusion of terms involving v\ v\ &e. This value is 
/— — * 1 A 2 a 2 v 2 
a - ' a ” + 2 a -vT 0 ’ 
for which we may write u=f J r gv 2 . 
We shall have a similar expression for the second medium, 
a ,=fi J rg, v2 - 
Consequently 
[Jj ftfl~ 
which again may be represented approximately under the form 13-j-CV 2 , 13 and C being- 
two constants which can only be known by experiment; but I shall adhere to the 
other more significant form. 
Hence we obtain for the refractive index of rays, which have undergone partial 
absorption, 
sin 0 fx. _ 1 g{f+ff v ‘ z ) 
sin 0 ; 1 + s 1 + s §,(/,+$/)’ 
whence sin 6.= - sin A— (1 -f s) sin 6. 
h ?{f+9n 
34. Though the expressions for p'pfq, (see No. 22 and 24) remain unchanged in 
form, their values will of course be affected by the change of 0 P which they involve. 
It would be needless to write their expressions over again, but it may be convenient 
for the sake of photometrical comparison to exhibit the ratios of vires vivce 
y Pi q[ £/ 5 
P P 9 9 
which are immediately derivable from them. They are as follows: — 
jJ sin 2 (0 — 0/) 
p sin 2 (0-j-0 ; ) 
Pj 1 sin 20. sin 25, 
P 1+s sin 2 (0 + 0 ; ) 
