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Now for air at 0° C. and 760 mm. pressure p = 1 '000292, so that
, = '000292 (p-p0)/Po = - '000146 (g2-Z72)/F2..........(14)
000146U* /« d fq2\ , * =-------FT-"]. ayU/i)^' ..................<15>
and the integral is the quantity already calculated under (9). It will be-observed that the effect is independent of the absolute size of the obstacle, and is proportional to the square of the fluid velocity U. As regards the sign of the effect, we see that, so long as (9) is positive, the retardation diminishes as y increases, and thus a ray originally parallel to x is bent inwards. This action in a range-finder leads to an over-estimate of the distance.
We may now consider the actual magnitude of %. If we take the highest value of (9), viz. T36, we have
X = - '000199 U*/V2............................(16)
To take a numerical example, let us suppose that
F(0° C., 760 mm.) = 33100 *. v ' sec.
The angle is then ~X= 2'28* x 10~7-
This is in circular measure, and it corresponds to '0470* second. An error of a twentieth part of a second may be considered negligible.
The corresponding problem with a sphere in place of the cylinder could be treated, but it is rather more complicated. It does not appear necessary to enter upon it here.
The objection may present itself that the motion in two dimensions round a cylinder does not well represent the case of a ship. This must be admitted; but it seems clear that a range-finder, carried well above the deck of a ship, would be much less liable to be disturbed in the manner discussed.
Addendum.In answer to a criticism the author wrote:
There may be different opinions as to how far conclusions for an infinite cylinder throw light on the case of the ship. Certainly my own impression is that the cutting away of the further parts of the cylinder would diminish the disturbance under discussion.
[* Apparently we should take V= 28,000 cm. per sec. (see footnote * on p. 600). We then find -X=3-18xlO"7in circular measure =-0656 second, or about one-fifteenth of a second. W.F.S.] should be from the full red of C to the blue-green of F, and this is just about what is observed. The agreement must be admitted to be a strong argument in favour of the theory. So far as I have seen, so great a range cannot be found in the surface-colour of any dye, even with the aid of polarized light.