30 



NA TV RE 



[November 9, 1899 



since this equation does not contain <p, p is constant whatever 

 the vahie of (p may be ; hence the bright parts of the field 

 arecircuhr rings surrounding point of contact of the plane and 

 spherical surface. 



The radii of the dark rings can be deduced from the relation 



2;-(i -cos 6) tan t- 



. 2« - I a . 

 2 cos o ' 



and from these equations it is easily shown that for the bright 

 rings 



and for dark rings 



tan t cos o 



corresponding values for 



This may be compared with 

 Newton's rings. 



In both cases the radii of the bright and dark rings vary as the 

 square roots of the even and odd numbers, and as the square 

 root of the radius of the sphere and the wave length (which is 

 analogous to a in the present case), but here the likeness 

 ceases. 



The rings here considered diminish as i increases, and 

 increase as a diminishes. 



Of course, in Newton's rings there is nothing which answers 

 to the angle a. 



The easiest way of examining these rings is to mould a small 

 circle of wire gauze to form part of a sphere (which can 

 readily be done by pressing it with a ball against any yielding 

 substance) and laying it, convex surface downwards, on a piece 

 of looking glass. 



In general two sets of rings will be seen, one due to the 

 wires of the warp, and the other of the woof of the gauze. 



When the eye however, looks parallel to one set of wires, 

 the rings of that set are all infinite, and only the set due to the 

 wires at right angles to the line of sight are visible. 



If the gauze is made to turn slowly on the point of contact, 

 both series appears, one growing, and the other diminishing, 

 which are exactly superposed when a = 45°. 



A curious effect may be observed when a thick plate of glass 

 is placed between the gauze and the looking glass. 



The rings in this case become coloured, showing blue on their 

 inner, and red on the outer margins of the dark bands. 



The explanation is obvious, for the pencils of white light 

 entering through the meshes of the gauze are dispersed on 

 entering the glass, and in the neighbourhood of the dark rings 

 only part of the dispersed pencils are cut off on their second 

 passage through the gauze, so that the light which reaches the 

 eye is coloured. 



If i is the thickness of the glass plate, the greatest colour 

 efifect is obtained when i = d/j./2clij. tan r, where l) is the diameter 

 of the wire and r the angle of refraction in the glass. 



When the glass plate is used, of course the smallest visible 

 ring is not that for which «=i, but it is unnecessary here to 

 enter on the alteration in the formula for p caused by putting 

 2r(i - cos 0)+/) for 2 r (i -cos 6). 



(2) Interference rings caused by two series of straight lines, 

 radiating at equal angles, from two centres in the same or 

 parallel planes. 



Let there be n lines in each series, then the angle between 

 successive lines in each series is 2ir/n. 



Let the lines of the first series be numbered i, 2, 3 . . . 

 p . . . n and those of the second i', 2', 2>' . • • 1' • • • '^> ^^^ 

 let the line i be parallel to the line i'. Then the angle made 

 by any line/ of the first series with another q' of the second is 

 (/ - q)2-irln, hence the intersections of all pairs of lines for which 

 (/ - q) is the same will lie on a circle passing through the two 

 centres and having this segmental angle. 



When the distance between the centres is a, the radius of the 

 circle is 



p = a cosec 2ir(p- q)ln 



if both centres are in the same plane, or 



{a + b sin i) cosec 2ir(/> - q)ln 

 if in different planes, where a = distance between the normals 

 to the planes through the centres, d the distance between the 

 planes, and i the angle made by the line of sight with the 

 normal. 



NO. 1567, VOL. 61] 



The loci of the intersections appear brighter than any other 

 part of the field of view, hence the intersections of the two' 

 series show as a family of bright and dark circles which all pass 

 through the two centres, and whose radii are as the cosecants 



of the multiples of — 

 «• 



This is shown in Fig. I. 



A pair of wheels of a carriage, one viewed through the other, 

 show the phenomenon very well, especially when the wheels 

 are turning fast enough to make the individual spokes indistinct. 



Under favourable circumstances as lo light and background, 

 the appearance of the rings, contracting and expanding as the 

 angle of view changes, is very striking. 



(3) Interference curves from two series of straight lines, one! 

 radiating and the other parallel. 



From a point P in the axis of Y let radiating lines be drawn 

 to cut the axis of X at equal intervals a, and at a, 2a, 3a, &c., 

 let lines be drawn parallel to Y. 



Then, if the distance of P from the origin is A, to determine 

 the coordinates of the intersection of the «th parallel line with 

 the M+/th radiating line, we have, since x = na, 



hence 



-''J^ {n+/>) a-na. 



The locus, therefore, obtained by giving the value o, 1,2.. 

 ot to M will be a series of points on a rectangular hyperbola 

 passing through P with its centre a.\.y = o x= -pa. 



Thus the field of view will show a family of hyperlolae (one for 

 each value of /), all passing through P, the parameters being 

 i^Jpha. . . 



In the same way, for the intersections of the «th parallel with 

 the 2« +/th radiating line we have 



■?^ (2 « + /) = /, 

 n 



which indicates a second fami'y of hyperbolae, the coordinates 



of the centres being - and -2. 

 2 2 



Similar families are formed by the intersection of the Mth 

 parallel, with the 3« + /th . . . \n + /th . . . &c., radiating 



lines, the corresponding centres beings = — , . . .-^, &c. 



3 4 



andA:= -^, -A&c. 

 3 4 



It will be readily seen that the dark and bright bands formed 

 by the interfering lines follow the short diagonals of the quadri- 

 laterals into which the two series of lines divide the field, and 

 that for the bands to be conspicuous, there should be a great 

 difference in the length of the two diagonals, and only a small 

 difference in the length of the sides of the quadrilaterals. 



For this reason only a part of each hyperbolic family is 

 recognisable. 



