On the Formula for the Dispersion of Light. 205 
Now, to compare this with the approximate development 
(2.) we may assume 
H, = vS(H?) 
_ 7, /S (He!) 
eee. vie 
which being substituted in (2.) will give 
ry w/e ope era 2/2" \*[S.(H?Ae*) |? 
a= 8(H)— 3(5) SHA) + GIS) “song 
== 18 &e. (6.) 
Hence it is manifest that by our assumption of H, and @, 
in the approximate formula, the two first terms of both deve- 
lopments are identical; but the third and subsequent terms 
will differ. We thus obtain an idea of the degree in which 
the approximation deviates from the truth, 
In the next place, adopting the exact development (4.) we 
may proceed to the important discussion of a method of de- 
termining the coefficients, and of actually computing the 
value of »% in any given case. The development may be ex- 
pressed in the following form, writing single letters for the 
coefficients : 
1 Fiih ee S gts 
== A,— Ai(=) + Ao(—] —, &e. (7.) 
Now, if r be the time of a vibration, or, what is the same 
thing, of the propagation of a wave, « being the reciprocal of 
the velocity, we shall have 
pt ay 
i eae 
Thus the series (7.) becomes 
! oan we wy" 
ap = Ao Ai), +4 (2) BAe (8.) 
Again, we might substitute other letters for the coefficients 
by making them include the powers of » so as to express the 
4, 1 : 
series in powers of tee only. By extracting the root, and 
developing the reciprocal of the polynomial, it will be easily 
seen that the series resulting will be still one of the same 
powers of (=), and may be expressed in this form, 
9 
= a + a, (=) + a, (on +, &c. (9.) 
