Between these it is possible to eliminate the three medium-constants a, a, a, and thus to deduce a general relation, valid for all media, between the four indices B D F MH, and the four periodic times TB TD TF TH

Now, a little consideration will show that this relation can only involve the two ratios of the three differences of the four indices, and the two ratios of the three differences of the four reciprocals of the squares of the periodic times. Taking these we may for abridgement write,

Then it will appear that the result of elimination will be a relation between the four quantities sp SF tp t only; and will not involve the four other quantities B HTB TH, if we previously substitute for PDF TD2 TF their respective expressions deduced from (14.) and (15.), which are

-2

This result of elimination will therefore be the same as if in the equations (13.) we had supposed

that is, the same as if we eliminated any two new quantities b and c between the three new equations

This last elimination is easy, and gives as the relation sought the following:

This relation may be expanded by substituting the values (14.) and (15.) so as to put it under the form,

and in this way the relation (20.) may be verified, as it will then be found to be satisfied independently of the three medium-constants a, a, a, by the expression (13.) for the four indices.

Now, to proceed to the actual calculation, we have Fraunhofer's values of a for the standard rays; these are obtained from interference, and are absolutely independent of any medium. Now if the time which light takes to traverse a given length l' in vacuo, will obviously have

If then we take t' as the unit of time, we have for the time of a vibration in vacuo

' =

Thus if Too inch, since by Fraunhofer's observations we have λ=00002451 inch, it follows that we have TB=2451, and similarly

TF= = '1794 ΤΗ = .1464.

(22.)

TD = 2175 Now, there is a circumstance which may be remarked among these numbers, which affords a considerable facility in our calculation. The square of T will be found to be almost exactly an harmonic mean between the square of the extreme values TB TH or we have

Availing ourselves of this circumstance we may put the relation (20.) or (21.) under the simpler form

4. StD (1-D)-SD = td (1—2tp)

(25.)

which are wholly functions of the values of T, viz.

Thus employing the values (22.) of TB TD THI the following numbers result:

log (-α) = 1.80441

log (-b) = 1.06281

Now, to take an example of a particular medium; for flint glass, No. 13, Fraunhofer found

MB = 1.6277

HF 16483

*H= 1671.

Hence by (26.) and the above logarithms, we may calculate the value of which will be found to result

HD = 1.63492;

by Fraunhofer's observation it was

D= 1.6350.

Such is the method of Sir W. R. Hamilton: he has, however, not only calculated this example, but has gone through the values of the index, for the same ray D in all the media examined by Fraunhofer. These results I will subjoin, adding a column of the same values as computed by myself, by a tentative method with only the approximate formula, from my paper in the Philosophical Transactions.

Third Series. Vol. 8. No. 46. March 1836. 2 A

I shall not here enter on any detailed remarks or comparisons of the results exhibited in this table. From it the reader will be enabled to form a correct judgement of the degree in which the approximate method is comparable with the exact; at least for media of no higher dispersive power than those examined by Fraunhofer. Meanwhile we may just observe, that the results here given by the exact formula are invariably a little in defect compared with those of observation; whereas the approximate numbers are sometimes in defect and sometimes in excess. To this circumstance, and some further investigations connected with it, I shall recur in a future communication.

In the last Number (p. 113) I alluded to the calculations of M. Rudberg. It may be worth while to observe that such a formula as that which he adopted empirically, may give results nearly coinciding with those of the formula derived from theory which I have used, as will appear by the following considerations.

M. Rudberg's formula, in my notation, becomes

1

μ

= a λ(m-1).

=a'), let us suppose a quantity H' so

a' 28
H' =
A-1'

a' λ = H' ( 1 − — ),

especially if we confine ourselves to the first two terms,

(which is usually sufficient,) making (m—1) = 6'

XLI. Remarks on a Note on a Pamphlet entitled "Newton and Flamsteed" in No. CX. of the Quarterly Review. By the Rev. W. WHEWELL, M.A. F.R.S., Fellow and Tutor of Trinity College, Cambridge.*

I

To the Editor of the Quarterly Review.

My dear Sir,

Trinity College, Cambridge, Feb. 3, 1836. HAVE just seen No. 110 of the Review; and I perceive that the reviewer of Mr. Baily's account of Flamsteed, in No. 109, has done my remarks on his article the honour of writing a note respecting them, which you have inserted.

As I do not see in this note any new arguments on the reviewer's side of our controversy, I do not conceive that I have occasion to add much to what I have already said, for I presume your readers do not look for an answer to mere hard words. A few additional remarks will, I think, enable competent judges to decide between us.

I asserted, and assert, that Flamsteed never fully comprehended or accepted Newton's theory ;-never understood the difference between the Newtonian theory of the causes of the celestial motions, and the empirical laws of phænomena which he himself called theories;-in short, the difference between a formula and an explanation-between the discovery of what occurred, and the discovery why it occurred-between an observer and a philosopher. I quoted a letter which proved this; nor does the reviewer venture to deny the clear inference which irresistibly follows from this quotation. But he takes refuge in "the whole tenour of the correspondence," without quoting a single passage. To any one capable of understanding the distinction which I have pointed out, the whole tenour of the correspondence shows Flamsteed to have had no glimpse of this difference. For example, he says (Account of Flam* From the 2nd edition of Mr. Whewell's Pamphlet. See our last Number, p. 139-147.