206 ~—— Prof. Powell’s Formula for Dispersion of Light 
Now, if we substitute this value of « in each term of the 
following expression derived from (8), viz. 
pe po 
O= —1+ Apwi— A, + Ag - &e. (10.) 
we shall have 
(=I + Ag [ao + a, t*? + &e.]? 
| + Av fiag +.a,77? +,&c.)*.077 \, , (11.) 
(VS 
| + Ag [ao +4, 77? + &e.]®°. r* 
e &c. 
If in this expression we collect the coefficients of the same 
powers of r and equate them respectively to zero, we shall at 
length obtain, 
= 
4% = ve - 
a, =iA,A,? (12.) 
Cay A, Ae a A, hare 
We might continue the process: but confining ourselves to 
the three first terms, we find, Ist, that a) a, are positive; and 
a, may probably be so; Zndly, that, since the first coeffi- 
cients A, A,, &c. are all independent of u, and since we have 
values of the second set a a, in terms of these only, therefore 
these last are also zndependent of »%, that is, are constant for 
all rays in the same medium, but differ for different media. 
Thus in the series (9.) the value of » will differ only by the 
change in the factor 7, that is, for rays whose times of vibra- 
tion are different, or, in other words, for rays whose lengths of 
waves are different, that is, for the different primary rays. 
We shall thus obtain the means of calculating the value of » 
for each ray independently of the medium, supposing we may 
restrict ourselves to the three first terms. For we have the 
three constants a @, @. the same in the series for each ray ; 
thus if we take such serieses for any four rays we can eliminate 
the three constants. 
Let us then consider the four rays (in Fraunhofer’s nota- 
tion) B, D, F, H; for any one medium, we shall have this sy- 
stem of equations, viz. 
rt Q a5 
bp = A + 4, TR + GTR” ) 
_ 
Hp = A + a tH + Me | (13.) 
sp 
2 -2 ~4 
bp = 4 + TR + Age | 
Py = % + TH + a, TH J 
