600 DISTURBANCE OF A. RANGE-FINDER BY [439 long as / > 2F. At 60° this condition has ceased to hold, and the maximum value of (9) occurs when, r > c. We may write generally e /(*) F(B) (9) when r = c (9) max. 0° 0 0 __ _ 10 •6840 •1404 •5436 — 20 1-2856 •2859 j -9997 — 30 1-7320 •4419 1-2901 — 36 1-9021 •5436 1-3585 — 40 1-9292 •6159 1-3133 — 50 1-9696 •8173 1-1523 ; — 60 1-7320 1-0605 •6715 •7072 70 1-2855 1-3680 - -0825 — . The maximum of (9) occurs when c8/r-2 =f/2F, and the maximum value is /2/4F. For 6 = 60°, the maximum is '7072. When 6 = 70°, (9) itself has changed sign, the transition occurring when F~f. For our present purpose we may take the highest value of (9) as 1'36. The general dynamical equation connecting pressure (p) and density (p) with velocity (q) in steady motion is .(11) It will suffice if we employ Boyle's law for the connexion of p and p, that is p = V2p, where V is the velocity of sound *. Thus Or, since the if p6 correspond with q = U, as at a distance from the cylinder. variations of density here contemplated are very small, Po Passing now to the optical side of the question, we have to consider the retardation experienced by a ray parallel to CD, due to the variable density. In accordance with a general principle, this when small enough may be calculated along the original ray, although the actual ray now follows a somewhat different course f. Thus if fyi be the change of refractive index /•co due to q, the retardation may be taken to be Spdsc, and the angle % J a through which the ray at P is turned is .(13) [* I.e. the " Newtonian" velocity of sound, =280 metres per second. W. F. S.] t For an application to the resolving power of prisms, reference may be made to Phil. Mug. Vol. vin. p. 269 (1879); Scientific Papers, Vol. i. p. 425. c. This state of things continues so-velocity U, is well known*. The problem may conveniently be reduced to one of "steady motion" by supposing the cylinder to be at rest while the fluid flows past it—a change which can make no