## Abstract

The paper presents two phases of the subject. (I) Schematic description of an apparently new type of optical instrument, designed by Liggett and Dodd, which permits conditions within wide limits for accommodation of the eyes independent of convergence, and also the measurement of muscular imbalance (strabismus) and range of fusion. It uses plane mirrors rather than refracting prisms, thus eliminating distortion, chromatic aberration and astigmatism in the images. The optical system is essentially different from that of the short-base range finder and of the Helmholtz stereoscope, in that (a) rays enter the eyes from the *outer* rather than the inner mirrors, and (b) the outer mirrors are variable in position through large angles. (II) Derivation of geometrical expressions for (1) magnitude of “field of view” (that is, angular range of visibility of secondary image of stationary point-object), (2) direction of field of view, and (3) boundaries of useful areas of mirrors, —applicable to a pair of independently variable plane mirrors with parallel axes in the respective planes, as adapted in an instrument of this kind (one pair of mirrors to each eye).

The designing entailed the fulfilling of several requirements having somewhat narrow limits and culminated in finding a suitable combination of some half-dozen variable quantities.

© 1930 Optical Society of America

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### Equations (14)

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(1)
$$\begin{array}{l}{\rho}^{2}=({{x}_{0}}^{2}+{{y}_{0}}^{2})+2d[{y}_{0}\hspace{0.17em}\text{sin}\hspace{0.17em}2\xi +2(d-{x}_{0})\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\xi ]\hfill \\ \varphi =\eta +\alpha +\u220a-\psi \hfill \end{array}\}$$
(2)
$${\sigma}_{3}={\sigma}_{2}-{\delta}_{2}=-\tau +2\eta +90.$$
(3)
$$\text{sin}\hspace{0.17em}\eta =\pm {\left(\frac{-B\pm ({B}^{2}-4AC{)}^{1/2}}{2A}\right)}^{1/2},$$
(4)
$${\sigma}_{3}=\theta +2\eta -2\xi +90,$$
(5)
$$\text{sin}\hspace{0.17em}\eta =\pm {\left(\frac{-B\pm ({B}^{2}+4C{)}^{1/2}}{-2}\right)}^{1/2},$$
(6)
$$\theta ={\sigma}_{3}-2\eta +2\xi -90,$$
(7)
$$2\eta =\varphi +\xi -\alpha +\psi ,$$
(8)
$$\text{sin}\hspace{0.17em}\varphi =-(b\hspace{0.17em}\text{cos}\hspace{0.17em}{\sigma}_{3}\pm \text{sin}\hspace{0.17em}{\sigma}_{3}(1-{b}^{2}{)}^{1/2}),$$
(9)
$$\text{sin}\hspace{0.17em}\psi =d[{y}_{0}\hspace{0.17em}\text{cos}\hspace{0.17em}2\xi +(d-{x}_{0})\hspace{0.17em}\text{sin}\hspace{0.17em}2\xi ]/[\rho [{(d-{x}_{0})}^{2}+{{y}_{0}}^{2}{]}^{1/2}],$$
(10)
$${\sigma}_{3}=\theta +2\eta -230\xb0,$$
(11)
$$\text{sin}\hspace{0.17em}\eta =\pm {\left(\frac{-B\pm {({B}^{2}+4C)}^{1/2}}{-2}\right)}^{1/2},$$
(12)
$$\theta ={\sigma}_{3}-2\eta +230\xb0,$$
(13)
$$2\eta =\varphi +\psi +50,$$
(14)
$$\text{sin}\hspace{0.17em}\varphi =-[b\hspace{0.17em}\text{cos}\hspace{0.17em}{\sigma}_{3}\pm \text{sin}\hspace{0.17em}{\sigma}_{3}(1-{b}^{2}{)}^{1/2}],$$