## Abstract

The present problem, in geometrical optics, deals with a more general case of inclined mirrors than the simpler and familiar ones of the kaleidoscope, the “hinged mirrors,” and the sextant. Here each of the two plane mirrors has its own independent axis in its own plane, the two axes being, however, parallel to each other, which requires that the intersection line of the two reflecting planes be always parallel to the two axes, but it may otherwise be located anywhere in space. The problem is stated so as to involve three parameters, in addition to the two independent variables. The main problem is that of the locus of the secondary image, O″, under these conditions. Derivations are given also for the locus of the primary image, O′, and for the intersection, C, of the mirror planes. From the geometrical standpoint the problem is that of two related isosceles triangles, OAO, OA0O″, where O is the object point and A, A0, are the points where the mirror axes cut the principal plane, all five points being coplanar. The problem is made more general physically by supposing that the mirrors are half-silvered. The geometrical situation is expressed also in terms of intersecting circles, useful for graphical construction.

© 1928 Optical Society of America

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1. From this it is obvious that any series of isosceles triangles related to each other by a tip of one of the two equal vertices of the first Δ in the series being coincident with the tip of one of the two equal vertices of the Δ next in the series, and the tip of the other equal vertex of this second Δ being in similar contact with a tip of one of the equal vertices of the third Δ, and so on, and the apex of each Δ being fixed in position, can represent the principal section of a geometrical system of reflecting planes, each capable of independent rotation about a fixed axis in its own plane, all the axes being mutually parallel, the tips of the two equal vertices of any one of the Δ’s representing point object and point image respectively for that mirror, the apex point being at its axis, and the image in that mirror becoming the object for the next mirror in the series. The principal section of the mirror itself is in each case the bisector of the apex angle.
2. If the three points O, A, A0, were to be located arbitrarily and perfectly generally in the xy plane, six parameters, the coordinates of the three points, would be required. Because the present restrictions place points A, A0, on the x-axis and at a specified distance apart, d, three of the six possible parameters are removed.
3. The number of images visible in the ordinary hinged mirrors, where the surfaces are regarded as reflecting clear to the vertex, is treated in a recent paper by H. Maurer, Physikalische Zs.,  29, No. 5, pp. 147–149; March1, 1928. The author cites two other papers on the subject. (See Science Abstracts, 1928, abstract no. 1540, also no. 1229.)

#### 1928 (1)

The number of images visible in the ordinary hinged mirrors, where the surfaces are regarded as reflecting clear to the vertex, is treated in a recent paper by H. Maurer, Physikalische Zs.,  29, No. 5, pp. 147–149; March1, 1928. The author cites two other papers on the subject. (See Science Abstracts, 1928, abstract no. 1540, also no. 1229.)

#### Maurer, H.

The number of images visible in the ordinary hinged mirrors, where the surfaces are regarded as reflecting clear to the vertex, is treated in a recent paper by H. Maurer, Physikalische Zs.,  29, No. 5, pp. 147–149; March1, 1928. The author cites two other papers on the subject. (See Science Abstracts, 1928, abstract no. 1540, also no. 1229.)

#### Physikalische Zs. (1)

The number of images visible in the ordinary hinged mirrors, where the surfaces are regarded as reflecting clear to the vertex, is treated in a recent paper by H. Maurer, Physikalische Zs.,  29, No. 5, pp. 147–149; March1, 1928. The author cites two other papers on the subject. (See Science Abstracts, 1928, abstract no. 1540, also no. 1229.)

#### Other (2)

From this it is obvious that any series of isosceles triangles related to each other by a tip of one of the two equal vertices of the first Δ in the series being coincident with the tip of one of the two equal vertices of the Δ next in the series, and the tip of the other equal vertex of this second Δ being in similar contact with a tip of one of the equal vertices of the third Δ, and so on, and the apex of each Δ being fixed in position, can represent the principal section of a geometrical system of reflecting planes, each capable of independent rotation about a fixed axis in its own plane, all the axes being mutually parallel, the tips of the two equal vertices of any one of the Δ’s representing point object and point image respectively for that mirror, the apex point being at its axis, and the image in that mirror becoming the object for the next mirror in the series. The principal section of the mirror itself is in each case the bisector of the apex angle.

If the three points O, A, A0, were to be located arbitrarily and perfectly generally in the xy plane, six parameters, the coordinates of the three points, would be required. Because the present restrictions place points A, A0, on the x-axis and at a specified distance apart, d, three of the six possible parameters are removed.

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### Figures (9)

F. 1

Physical situation of inclined mirrors in the present problem.

F. 2

The geometrical problem of two related isoscles triangles.

F. 3

The geometrical problem in detail.

F. 4

Geometrical situation, showing the intermediate triangle, OOA0, used in the derivations.

F. 5

Particular case: ξ = π/2; η = 60° approximately.

F. 6

Particular case: ξ = π/2; η = 60°.

F. 7

Particular case: ξ = 0; η = 30°.

F. 8

Particular case: η = π/2; ξ = 20°.

F. 9

The geometrical situation in terms of intersecting circles.

### Equations (20)

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$cos λ = ( d 2 + D 2 − r 2 ) / ( 2 D d ) ; cos μ = ( r 2 + d 2 − D 2 ) / ( 2 r d ) ; cos ν = ( r 2 + D 2 − d 2 ) / ( 2 r D ) ;$
$cos λ = x 1 / ( x 1 2 + y 1 2 ) 1 / 2 ; cos μ = ( d − x 1 ) / [ ( d − x 1 ) 2 + y 1 2 ] 1 / 2 ; cos ν = [ ( x 1 2 + y 1 2 ) − x 1 d ] / { ( x 1 2 + y 1 2 ) · [ ( d − x 1 ) 2 + y 1 2 ] } 1 / 2 ,$
$ρ 2 = D 2 + 4 r sin α [ r sin α + D sin ( ν − α ) ] ,$
$η − β = η − ∊ − α + ψ = ξ + ψ − α$
$ϕ = η + β = η + ∊ + α − ψ = 2 η − ξ + α − ψ .$
$x 2 = d + r cos ( ξ − α ) , y 2 = r sin ( ξ − α ) .$
$cos α = cos [ π − ( μ + ξ ) ] = − cos ( μ + ξ ) = ( sin μ sin ξ − cos μ cos ξ ) = [ y 1 sin ξ − ( d − x 1 ) cos ξ ] / [ ( d − x 1 ) 2 + y 1 2 ] 1 / 2 ,$
$sin α = [ 1 − cos 2 α ] 1 / 2 = [ y 1 cos ξ + ( d − x 1 ) sin ξ ] / [ ( d − x 1 ) 2 + y 1 2 ] 1 / 2$
$α = π − μ − ξ = π − ξ − cos − 1 { ( d − x 1 ) / [ ( d − x 1 ) 2 + y 1 2 ] 1 / 2 } ,$
$x 2 = d + y 1 sin 2 ξ − ( d − x 1 ) cos 2 ξ ,$
$y 2 = − [ y 1 cos 2 ξ + ( d − x 1 ) sin 2 ξ ] .$
$x 0 = ρ 1 cos ( π + η ) = − ρ 1 cos η ,$
$y 0 = ρ 1 sin ( π + η ) = − ρ 1 sin η ,$
$ρ 2 = x 2 2 + y 2 2 = ( x 1 2 + y 1 2 ) + 2 d [ y 1 sin 2 ξ + 2 ( d − x 1 ) sin 2 ξ ] ,$
$ϕ = η + β ,$
$sin ψ = [ d sin ( α − ξ ) ] / ρ = d [ y 1 cos 2 ξ + ( d − x 1 ) sin 2 ξ ] / [ ρ [ ( d − x 1 ) 2 + y 1 2 ] 1 / 2 ]$
$R 2 = ρ 2 + ρ 1 2 − 2 ρ ρ 1 cos [ ϕ − ( π + η ) ] ,$
$R 2 = ρ 2 + ( d sin ξ ) 2 / sin 2 ∊ + 2 ρ d cos ( ϕ − η ) sin ξ / sin ∊ .$
$R 2 = C O 2 ¯ = ρ 1 2 + D 2 + 2 ρ 1 D cos ( λ − η ) .$
$R 2 = { D 2 + ( d sin ξ / sin ∊ ) 2 + [ ( 2 D d sin ξ ) / sin ∊ ] · [ ( D 2 + d 2 − r 2 2 D d ) cos η + sin cos − 1 + ( D 2 + d 2 − r 2 2 D d ) sin η ] } .$