598 DISTURBANCE OF A RANGE-FINDER BY [439 difference to the refraction. The velocity-potential is then expressed in polar coordinates by .......................(1) •(2) 0 = ft (r + H cos 9, \ Tj where r is measured from the centre ; whence c2 Lt/UJ TT / -, L> \ n •f=U(l- - costf, dr \ rV Here d(f>/dr vanishes when r = c at the surface of the cylinder, and at a great distance the resultant velocity is U, parallel to x... In general As will be stated more at length presently, the reduction of pressure and density is proportional to g2 — U2, so that the (abnormal) optical retardation of a ray parallel to as is proportional to / (q2 — Uz) dx, in which the upper limit for as may .be treated as infinite. The difference of retardations for two infinitely near rays parallel to a>, divided by the distance between them, gives the angle of refraction, which is thus represented by -r- \q*dx, or by since the limits of x are the same for both. Now and, since so that dy \r drjdy = y/r, d ( « ' dy Vr r6 dy In the integration with respect to so, y is constant, say j3, and if «, /3 be the rectangular coordinates of the point P in the figure at which the refraction is to be estimated, the limits for so are a and oo . Thus - di: — - 4>&8 (fi*-4- 2 *c P(P + = 6, OQ or OP-r.ical and prismatic dispersion or of mirage. It is doubtless the fact that in the rear of the disturbing body the motion differs greatly from that assumed; but in front of it the difference is much less, and not such as to nullify the conclusions that may be drawn. The first step is accordingly to specify the motion, and to determine the square of the total velocity (q) on which depends the reduction of pressure.