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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 26,
- Issue 8,
- pp. 323-337
- (1936)
- •doi: 10.1364/JOSA.26.000323

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- Cf. Bibliography.

- Under the shape category, aniseikonia also includes declination errors. Such errors can usually be interpreted in terms of a meridional magnification of the dioptric image at some oblique axis.

- Zero verging power magnification lenses.

- The term “retinal image” as used here, describes that section of the bundle of image forming rays in the vitreous humor of the eye which falls upon the retina, and to which the retinal elements respond.

- The sign system used is as follows: light is incident from the left; distances are measured from surfaces; distances to left are negative; distances to right are positive. All separations for purposes of development are taken negative. A reduced distance is an actual distance in an optical medium divided by the index of refraction of that medium.

- Surface powers are defined by D=(n−1)/r, where r is the radius of the curve in meters and n is the index of refraction.

- Cf. T. Smith, “The Primordial Coefficients of Asymmetrical Lenses,” Trans. Opt. Soc. 29, 170 (1928). M. Herzberger, Strahlenoptik, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 35, 61 (1931). They may be determined by other methods, e.g., cf. C. Pendlebury, Lenses and Systems of Lenses (London, 1884). J. G. Leatham, Symmetrical Optical Instrument (Cambridge, 1908). A has the dimension of a reciprocal of length (since a surface power is the reciprocal of a distance); B and C have no dimensions, being numbers; and D has the dimension of length. These coefficients are related by BC−AD=1.

[Crossref] - If such a lens corrects an ametropic eye, for a given object distance, V is the reciprocal of the distance measured from the ocular surface of the lens to the plane that is conjugate to the retina of the ametropic eye.

- Though this division of the magnification into factors for the eye and correcting lens is the most straightforward from the point of view of geometrical optics, it is not the only one, as will be shown later.

- Here t, c, h and u are taken positive.

- This factor is designated the “effectivity factor,” by T. Smith. Cf. Trans. Opt. Soc. 26, 31 (1924).

[Crossref] - These relations, easily found by applying the Gauss theory to the object and image distances referred to a second pair of conjugate points, are identical with the vergence equations of A. Gullstrand. Cf. Helmholtz, Physiological Optics, Eng. Trans., Vol. I, p. 277ff.

- The entrance pupil of the eye is the image of the real pupil formed by that part of the dioptric system which lies between the real pupil and the object. The distance of the entrance pupil from the pole of the cornea can be measured by means of a corneal microscope. The exit pupil will be the image of the real pupil formed by that part of the dioptric system lying between the true pupil and the retina.

- This is essentially correct as concerns the human eye—see unpublished work, Determination of the Effective Entrance Pupil, by K. N. Ogle, October, 1933.

- This may not be strictly true in cases of large spherical aberration. It must be borne in mind that this discussion pertains to centered systems. Actually the pupil is decentered and the dioptric system of the eye is both decentered and tipped relative to visual axis.

- This relation differs from that of Eq. (19) in that he here is measured from the entrance pupil instead of the cornea. In the schematic eye the entrance pupil lies about 3.2 mm behind the cornea. Because this expression is best adapted for either blurred or sharp imagery it is perhaps the best to use empirically.

- If Vo=0, D1=0, Lo=angular magnification of plane parallel of thickness t mm.

- Comparing the ocular images in any other manner explicitly involves the absolute magnitude of one of the k’s. The ratio permits one to compare the sizes in percentages.

- As would be the case in the testing instrument.

- The Gauss coefficients for a four surface system areA=A12(1+D4c2)+A34(1+D1c1)+A12A34w,A12=D1+D2+D1D2c1,A34=D3+D4+D3D4c2,B=(1+D2c1)(1+D4c2)+(c1+w(1+D2c1))A34,C=c2A12+(1+D1c1)(1+D3c2)+A12w(1+D3c2),D=(1+D2c1)c2+c1(1+D3c2)+(1+D3c2)(1+D2c1)w.

- Cf. T. Smith, “Back Vertex Powers,” Trans. Opt. Soc., 26, 35 (1924), for case when U=0, i.e., parallel incident light.

- Taking c, h and u as positive here.

- In general R differs from unity by a few percent, and it has been convenient to express R as R=1+e, and refer to the percent eikonic difference, viz., 100e.

- Referred to the points before the eyes at which the vergence powers of the trial case lenses were specified.

- These relations follow since the positions of the images formed by the spectacle lenses must be the same as were the positions of the images formed by the corresponding trial case lenses.

- This additive property is found in most of the trial case sets manufactured today.

- The L factor is unity for distant vision and hence does not enter into the cylindrical excess magnification. If cylindrical lenses are included before both eyes in the test, even though one is of zero power, the L factor drops out of the eikonic trial case lens ratio.

- One means for doing this is to design the shape of the cylindrical lens for distant vision so that its shape factor So1 just offsets the T factor, i.e., So1T=1. A different set of cylinders would be required for near vision unless cylindrical lenses are always placed before both eyes (though one may be of zero power).

- In terms of the surface powers and thickness of the lens the vertex power (cf. Eq. (19)) can be written Vo=D1+D2+e, where e is an allowance factor equal to SoD12c(=D12c, approximately) that can be found from prepared tables.

- Dr. E. D. Tillyer first pointed out this simplification.

- These lenses have excellent field properties with negligible distortion.

- This follows from the series expansion of logeM, i.e.,loge M=(M-1)-12(M-1)2+13(M-1)3⋯.When 12(M-1)2 is below precision of measurements, (100) logeM is identical with percent magnification.

Cf. T. Smith, “The Primordial Coefficients of Asymmetrical Lenses,” Trans. Opt. Soc. 29, 170 (1928). M. Herzberger, Strahlenoptik, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 35, 61 (1931). They may be determined by other methods, e.g., cf. C. Pendlebury, Lenses and Systems of Lenses (London, 1884). J. G. Leatham, Symmetrical Optical Instrument (Cambridge, 1908). A has the dimension of a reciprocal of length (since a surface power is the reciprocal of a distance); B and C have no dimensions, being numbers; and D has the dimension of length. These coefficients are related by BC−AD=1.

[Crossref]

This factor is designated the “effectivity factor,” by T. Smith. Cf. Trans. Opt. Soc. 26, 31 (1924).

[Crossref]

Cf. T. Smith, “Back Vertex Powers,” Trans. Opt. Soc., 26, 35 (1924), for case when U=0, i.e., parallel incident light.

These relations, easily found by applying the Gauss theory to the object and image distances referred to a second pair of conjugate points, are identical with the vergence equations of A. Gullstrand. Cf. Helmholtz, Physiological Optics, Eng. Trans., Vol. I, p. 277ff.

This is essentially correct as concerns the human eye—see unpublished work, Determination of the Effective Entrance Pupil, by K. N. Ogle, October, 1933.

Cf. T. Smith, “The Primordial Coefficients of Asymmetrical Lenses,” Trans. Opt. Soc. 29, 170 (1928). M. Herzberger, Strahlenoptik, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 35, 61 (1931). They may be determined by other methods, e.g., cf. C. Pendlebury, Lenses and Systems of Lenses (London, 1884). J. G. Leatham, Symmetrical Optical Instrument (Cambridge, 1908). A has the dimension of a reciprocal of length (since a surface power is the reciprocal of a distance); B and C have no dimensions, being numbers; and D has the dimension of length. These coefficients are related by BC−AD=1.

[Crossref]

This factor is designated the “effectivity factor,” by T. Smith. Cf. Trans. Opt. Soc. 26, 31 (1924).

[Crossref]

Cf. T. Smith, “Back Vertex Powers,” Trans. Opt. Soc., 26, 35 (1924), for case when U=0, i.e., parallel incident light.

[Crossref]

This factor is designated the “effectivity factor,” by T. Smith. Cf. Trans. Opt. Soc. 26, 31 (1924).

[Crossref]

Cf. T. Smith, “Back Vertex Powers,” Trans. Opt. Soc., 26, 35 (1924), for case when U=0, i.e., parallel incident light.

Taking c, h and u as positive here.

In general R differs from unity by a few percent, and it has been convenient to express R as R=1+e, and refer to the percent eikonic difference, viz., 100e.

Referred to the points before the eyes at which the vergence powers of the trial case lenses were specified.

These relations follow since the positions of the images formed by the spectacle lenses must be the same as were the positions of the images formed by the corresponding trial case lenses.

This additive property is found in most of the trial case sets manufactured today.

The L factor is unity for distant vision and hence does not enter into the cylindrical excess magnification. If cylindrical lenses are included before both eyes in the test, even though one is of zero power, the L factor drops out of the eikonic trial case lens ratio.

One means for doing this is to design the shape of the cylindrical lens for distant vision so that its shape factor So1 just offsets the T factor, i.e., So1T=1. A different set of cylinders would be required for near vision unless cylindrical lenses are always placed before both eyes (though one may be of zero power).

In terms of the surface powers and thickness of the lens the vertex power (cf. Eq. (19)) can be written Vo=D1+D2+e, where e is an allowance factor equal to SoD12c(=D12c, approximately) that can be found from prepared tables.

Dr. E. D. Tillyer first pointed out this simplification.

These lenses have excellent field properties with negligible distortion.

This follows from the series expansion of logeM, i.e.,loge M=(M-1)-12(M-1)2+13(M-1)3⋯.When 12(M-1)2 is below precision of measurements, (100) logeM is identical with percent magnification.

These relations, easily found by applying the Gauss theory to the object and image distances referred to a second pair of conjugate points, are identical with the vergence equations of A. Gullstrand. Cf. Helmholtz, Physiological Optics, Eng. Trans., Vol. I, p. 277ff.

The entrance pupil of the eye is the image of the real pupil formed by that part of the dioptric system which lies between the real pupil and the object. The distance of the entrance pupil from the pole of the cornea can be measured by means of a corneal microscope. The exit pupil will be the image of the real pupil formed by that part of the dioptric system lying between the true pupil and the retina.

This is essentially correct as concerns the human eye—see unpublished work, Determination of the Effective Entrance Pupil, by K. N. Ogle, October, 1933.

This may not be strictly true in cases of large spherical aberration. It must be borne in mind that this discussion pertains to centered systems. Actually the pupil is decentered and the dioptric system of the eye is both decentered and tipped relative to visual axis.

This relation differs from that of Eq. (19) in that he here is measured from the entrance pupil instead of the cornea. In the schematic eye the entrance pupil lies about 3.2 mm behind the cornea. Because this expression is best adapted for either blurred or sharp imagery it is perhaps the best to use empirically.

If Vo=0, D1=0, Lo=angular magnification of plane parallel of thickness t mm.

Comparing the ocular images in any other manner explicitly involves the absolute magnitude of one of the k’s. The ratio permits one to compare the sizes in percentages.

As would be the case in the testing instrument.

The Gauss coefficients for a four surface system areA=A12(1+D4c2)+A34(1+D1c1)+A12A34w,A12=D1+D2+D1D2c1,A34=D3+D4+D3D4c2,B=(1+D2c1)(1+D4c2)+(c1+w(1+D2c1))A34,C=c2A12+(1+D1c1)(1+D3c2)+A12w(1+D3c2),D=(1+D2c1)c2+c1(1+D3c2)+(1+D3c2)(1+D2c1)w.

If such a lens corrects an ametropic eye, for a given object distance, V is the reciprocal of the distance measured from the ocular surface of the lens to the plane that is conjugate to the retina of the ametropic eye.

Though this division of the magnification into factors for the eye and correcting lens is the most straightforward from the point of view of geometrical optics, it is not the only one, as will be shown later.

Here t, c, h and u are taken positive.

Cf. Bibliography.

Under the shape category, aniseikonia also includes declination errors. Such errors can usually be interpreted in terms of a meridional magnification of the dioptric image at some oblique axis.

Zero verging power magnification lenses.

The term “retinal image” as used here, describes that section of the bundle of image forming rays in the vitreous humor of the eye which falls upon the retina, and to which the retinal elements respond.

The sign system used is as follows: light is incident from the left; distances are measured from surfaces; distances to left are negative; distances to right are positive. All separations for purposes of development are taken negative. A reduced distance is an actual distance in an optical medium divided by the index of refraction of that medium.

Surface powers are defined by D=(n−1)/r, where r is the radius of the curve in meters and n is the index of refraction.

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Diagram used in study of retinal image size when correcting an ametropic eye with a single ophthalmic lens.

Diagram used in study of retinal image size with blurred optical imagery.

Schematic representation of an ophthalmic lens used before the eye showing notations used in text.

Diagram used in study of retinal image size when correcting an ametropic eye with two ophthalmic lenses.

Illustration of spectacles for the correction of aniseikonia using a fitover lens.

**Table I** Zero verging power lenses.Lenses which change the apparent size of the object without materially changing the vergence of the light which enters the eye. Lenses designed to have zero verging power for a visual distance of 75 cm.

Equations on this page are rendered with MathJax. Learn more.

$$Auv+Bv-Cu-D=0,$$

$$\left|\begin{array}{cc}B& D\\ A& C\end{array}\right|=\left|\begin{array}{cc}1& 0\\ {D}_{1}& 1\end{array}\right|\hspace{0.17em}\left|\begin{array}{cc}1& {c}_{1}\\ 0& 1\end{array}\right|\hspace{0.17em}\left|\begin{array}{cc}1& 0\\ {D}_{2}& 1\end{array}\right|\cdots \left|\begin{array}{cc}1& 0\\ {D}_{n}& 1\end{array}\right|.$$

$$m=1/(Au+B)$$

$${v}_{e}=(\mathbf{C}{u}_{e}+\mathbf{D})/(\mathbf{A}{u}_{e}+\mathbf{B})$$

$${m}_{e}=i/I=1/(\mathbf{A}{u}_{e}+\mathbf{B}),$$

$$v=(Cu+D)/(Au+B),$$

$$\left|\begin{array}{cc}B& D\\ A& C\end{array}\right|=\left|\begin{array}{cc}1& 0\\ {D}_{1}& 1\end{array}\right|\hspace{0.17em}\left|\begin{array}{cc}1& c\\ 0& 1\end{array}\right|\hspace{0.17em}\left|\begin{array}{cc}1& 0\\ {D}_{2}& 1\end{array}\right|,$$

$$\begin{array}{ll}\hfill A=& {D}_{1}+{D}_{2}+{D}_{1}{D}_{2}c,\\ \hfill B=& 1+{D}_{2}c,\\ \hfill C=& 1+{D}_{2}c,\\ \hfill D=& c,\\ \hfill \text{and}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}u=& p+h+t.\end{array}$$

$${m}_{L}=I/0=1/(Au+B).$$

$$v=\frac{1+{F}_{1}c}{{F}_{1}+{D}_{2}(1+{F}_{1}c)}.$$

$${V}^{8}=1/v={F}_{1}/(1+{F}_{1}c)+{D}_{2}.$$

$$S=1/(1+{F}_{1}c),$$

$$V=S{F}_{1}+{D}_{2}.$$

$${m}_{L}=SU/V.$$

$$\frac{i}{0}={m}_{L}{m}_{e}=\left[S\frac{U}{V}\right]\hspace{0.17em}\left[\frac{1}{\mathbf{A}{u}_{e}+\mathbf{B}}\right].$$

$${u}_{e}=v+h=(1+Vh)/V$$

$$\frac{i}{0}=\left[\frac{SU}{1+Vh}\right]\hspace{0.17em}\left[\frac{{u}_{e}}{\mathbf{A}{u}_{e}+\mathbf{B}}\right].$$

$$i/0=(SUP)(\mathbf{D}-\mathbf{B}{v}_{e}),$$

$$\frac{i}{0}=\left[S{U}_{p}P\right]\hspace{0.17em}\left[\frac{\mathbf{D}-\mathbf{B}{v}_{e}}{p}\right].$$

$$M=S{U}_{p}P$$

$$M={S}_{o}LP,$$

$$i/0=ME,$$

$${s}^{\prime}={m}^{2}s/(1+mAs)$$

$$\text{and}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{m}_{e}=m/(1+mAs),$$

$$s=(1/V)+{h}_{x},$$

$$i/0=({S}_{o}{L}_{x}{P}_{x})({s}^{\prime}/m{p}_{x}),$$

$$i/0={M}_{x}{E}_{x}.$$

$$i/0=({S}_{o}{L}_{e}{P}_{e})(y/m{p}_{e}),$$

$${i}_{b}/i={y}_{o}/y,$$

$$\begin{array}{ll}\hfill {i}_{b}/0=& ({S}_{o}{L}_{c}{P}_{e})({y}_{o}/m{p}_{e})\\ \hfill \text{or}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{i}_{b}/0=& {M}_{L}{E}_{b},\end{array}$$

$${M}_{e}={S}_{o}LP,$$

$$M={S}_{o}{P}_{o}{L}_{o},$$

$${i}^{\prime}=ki,$$

$$\frac{{{i}_{1}}^{\prime}}{{{i}_{2}}^{\prime}}=\frac{{k}_{1}{i}_{1}}{{k}_{2}{i}_{2}}=\left[a\right]\frac{{i}_{1}}{{i}_{2}},$$

$$i=MEO,$$

$$\begin{array}{ll}\hfill {i}_{1}=& {M}_{1}{E}_{1}O\\ \hfill \text{and}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{i}_{2}=& {M}_{2}{E}_{2}O.\end{array}$$

$$\frac{{i}_{1}}{{i}_{2}}=\left[\frac{{M}_{1}{E}_{1}}{{M}_{2}{E}_{2}}\right]=\left[\frac{{M}_{1}}{{M}_{2}}\right]\hspace{0.17em}\left[d\right],$$

$$\frac{{{i}_{1}}^{\prime}}{{{i}_{2}}^{\prime}}=\left[\frac{{M}_{1}}{{M}_{2}}\right]\hspace{0.17em}\left[d\right]\hspace{0.17em}\left[a\right].$$

$$i/0=(ME),$$

$$\left[\frac{{M}_{1}{E}_{1}}{{M}_{2}{E}_{2}}\right]$$

$${V}_{t}=(Au+B)/(Cu+D),$$

$${m}_{L}=I/0=1/(Au+B).$$

$${V}_{t}={V}_{o2}+\frac{{V}_{1}{{S}_{o2}}^{2}}{1+{V}_{1}(w+{S}_{o2}{c}_{2})},$$

$${m}_{L}=({S}_{1}{S}_{o2}T)(U/{V}_{t}),$$

$$1/T=1+{V}_{1}(w+{S}_{o2}{c}_{2}).$$

$$\frac{i}{0}={m}_{e}{m}_{L}=\left[{S}_{1}{S}_{o2}T\frac{U}{V}\right]\hspace{0.17em}\left[\frac{\mathbf{D}-\mathbf{B}{v}_{e}}{{u}_{e}}\right].$$

$${u}_{e}=(1+Vh)/V,$$

$$i/0=ME,$$

$$\text{where}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}M={S}_{o1}{S}_{o2}TLP,$$

$$\begin{array}{ll}\hfill {V}_{1}=& {S}_{1}({D}_{1}-U)+{D}_{2},\\ \hfill {S}_{1}=& \frac{1}{1-({D}_{1}-U){c}_{1}}=\frac{{S}_{o1}u}{u+{S}_{o1}{c}_{1}},\\ \hfill U=& 1/u,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}u=p-h-{t}_{1}-{t}_{2}-w,\\ \hfill {V}_{1}=& {V}_{o1}-\frac{{{S}_{o1}}^{2}}{u+{S}_{o1}{c}_{1}},\\ \hfill {V}_{o1}=& {S}_{o1}{D}_{1}+{D}_{2},\\ \hfill {V}_{o2}=& {S}_{o2}{D}_{3}+{D}_{4},\end{array}$$

$${R}_{o}={\left[\frac{{M}_{1}}{{M}_{2}}\right]}_{e}\left[d\right]\hspace{0.17em}\left[a\right].$$

$$\frac{{{i}_{1}}^{\prime}}{{{i}_{2}}^{\prime}}=R{\left[\frac{{M}_{1}}{{M}_{2}}\right]}_{e}\hspace{0.17em}\left[d\right]\hspace{0.17em}\left[a\right]=1,$$

$$R=1/{R}_{o}.$$

$$\frac{{{i}_{1}}^{\prime}}{{{i}_{2}}^{\prime}}={\left[\frac{{M}_{1}}{{M}_{2}}\right]}_{s}\hspace{0.17em}\left[d\right]\hspace{0.17em}\left[a\right]=1.$$

$${\left[\frac{{M}_{1}}{{M}_{2}}\right]}_{s}=R{\left[\frac{{M}_{1}}{{M}_{2}}\right]}_{e},$$

$${\frac{{({S}_{o}LP)}_{1}}{{({S}_{o}LP)}_{2}}]}_{s}=R{\frac{{({S}_{o1}{S}_{o2}TLP)}_{1}}{{({S}_{o1}{S}_{o2}TLP)}_{2}}]}_{e}.$$

$$1/P=1-Vh,$$

$${h}_{s}={h}_{e}+\mathrm{\Delta}h.$$

$${\frac{{({S}_{o}L{P}^{\prime})}_{1}}{{({S}_{o}L{P}^{\prime})}_{2}}]}_{s}={R\frac{{({S}_{o1}{S}_{o2}TL)}_{1}}{{({S}_{o1}{S}_{o2}TL)}_{2}}]}_{e},$$

$$\text{where}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{P}^{\prime}=\frac{1}{1-V(\mathrm{\Delta}h)},$$

$$p={p}_{o}+{h}_{e}.$$

$${M}_{a}={S}_{o}L{P}^{\prime},$$

$$\text{where}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{S}_{o}=1/(1-{D}_{1}c),$$

$$\begin{array}{l}{({V}_{1})}_{s}={({V}_{1})}_{e}/({{P}_{1}}^{\prime}),\\ {({V}_{2})}_{s}={({V}_{2})}_{e}/({{P}_{2}}^{\prime}).\end{array}$$

$${V}_{ot}={V}_{o2}+\frac{{V}_{o1}{{S}_{os}}^{2}}{1-{V}_{o1}(w+{S}_{os}{c}_{2})},$$

$${{V}_{o1}}^{\prime}=\frac{{V}_{o1}{{S}_{os}}^{2}}{1-{V}_{o1}(w+{S}_{os}{c}_{2})}.$$

$${V}_{ot}={V}_{o2}+{{V}_{o1}}^{\prime}.$$

$${\frac{{({S}_{o}{P}^{\prime})}_{1}}{{({S}_{o}{P}^{\prime})}_{2}}]}_{s}={R\frac{{N}_{1}}{{N}_{2}}]}_{c}$$

$$\begin{array}{ll}\hfill \text{and}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{({V}_{o})}_{1}{{]}_{s}}^{29}=& {V}_{o}{]}_{e}/{P}^{\prime}{]}_{1},\\ \hfill {({V}_{o})}_{2}{]}_{s}=& {V}_{o}{]}_{e}/{P}^{\prime}{]}_{2}.\end{array}$$

$${M}^{\prime}={S}_{o}L{P}^{\prime}/N.$$

$${{M}_{1}}^{\prime}/{{M}_{2}}^{\prime}{]}_{s}F=R,$$

$$\mathbf{V}=V+1/(p-h),$$

$$\begin{array}{ll}\hfill M=& {S}_{o}\frac{p-h}{p-h-t+{S}_{o}c},\\ \hfill {V}_{o}=& \frac{{{S}_{o}}^{2}}{u+{S}_{o}c}-\frac{1}{p-h},\\ \hfill \text{where}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{V}_{o}=& {S}_{o}{D}_{1}+{D}_{2}.\end{array}$$

$$\begin{array}{ll}\hfill {D}_{1}=& \frac{n((M-1)/M)(p-h)-(n-1)t}{t(p-h-t)}\\ \hfill \text{and}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{D}_{2}=& \frac{n(p-h-t)(M-1)-t(n-1)}{t(p-h)}.\end{array}$$

$$M={S}_{o}=1/(1-{D}_{1}t/n),$$

$$\begin{array}{lll}\text{whence}\hfill & \hfill {D}_{1}=& (M-1)n/Mt\\ \text{or}& \hfill M=& 1-{D}_{2}t/n,\\ \hspace{0.17em}& \hfill {D}_{2}=& -M{D}_{1}=-(M-1)n/t.\end{array}$$

$$h(M{S}_{o}-1)-(t/n)(n-{S}_{o})=0$$

$$t=\frac{nh({M}^{2}-1)(p-h)}{h({M}^{2}-1)+p(n-M)}.$$

$$LM=L{S}_{o}+LL+LP,$$

$$L{M}_{1}{]}_{s}-L{M}_{2}{]}_{s}=LR+L{M}_{1}{]}_{e}-L{M}_{2}{]}_{e}.$$

$${\text{log}}_{e}\hspace{0.17em}M=(M-1)-{\scriptstyle \frac{1}{2}}{(M-1)}^{2}+{\scriptstyle \frac{1}{3}}{(M-1)}^{3}\cdots .$$

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