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position is attained when the axis of rotation passes through the first nodal point, in which case the object, real or virtual, will be situated at the first principal focus.
The light will then emerge as a parallel pencil, and the image can be viewed through a telescope focussed for parallel rays, as in one of the well-known methods of determining the focal length of a thin lens. A further advantage of this position is that the nodal points are determined directly as in the ordinary nodal-slide method.
We will now apply these formula to the example given in Prof. Anderson's second paper (Phil. Mag. July 1917, p. 76),
OP1 =1 = 142 cm.; OP2 =y1 = 94 cm.; and m1 =
OP==29.1 cm.; OP'=y2=8.3 cm. ; and m2= d=113.8 cm.; H2H1=a=2·43 cm.
The distance H2O in Prof. Anderson's figure Phil. Mag. Jan. 1917, p. 158), in which the principal and nodal points are coincident, is given by
OP OP1-OP2 Ꮖ -y which is but a particular case of the general expression obtained by Prof. Anderson.
Hence in the first position H10=—
and in the second position H,O=
== -3.40 cm.
These positions in relation to the nodal points are shown to scale in fig. 2.
Suppose now that 0=5°=0·0873 radian and that the axis of rotation is moved 1 mm. from the nul position towards the object.
Then, in the first case, —— 2·60+0·1= 2.5 cm.; m1=0·0662; and the displacement of the image calculated by equation (1) is 0.0083 cm. In the second case we have 13·40+0.1 3.3 cm.; mq = 0-285 and the displacement is 0.0062 cm. If, however, the ordinary nodalslide method had been employed, in which case Ŏ would
coincide with H2, then l=−2·33, m=0, and the displacement under the same conditions would have been 0.0087 cm.
These small displacements may also be calculated more simply by using the approximate formula
In the first case l -2.6 and 8s=0.0082, in the second case 34 and 8s = 0.0062; while in the ordinary nodal-slide method = 2.43 and 8s=0.0087. - These results agree closely with those obtained by the more exact formula. Or, to put the matter rather differently, if we suppose the smallest observable displacement to be 0.01 cm., then the distances through which the axis would have to be moved from the nul position to produce this displacement would be 1.20, 161, and 1.15 mm. in the three cases respectively. These examples show that in this particular case the nul point can be determined with greater accuracy by the ordinary nodal-slide method than by the general method.
If, however, a convergent combination forming a real image, as in fig. 1, had been employed, the nul point would lie between N1 and N2, and its position could be determined with greater accuracy by the general method than by the ordinary nodal-slide method. In this case the nul point is indicated by Oo, which is the point of intersection of QQ2 with the optic axis.
Again, if, with the same convergent combination, a virtual image were formed, a case not likely to arise in the determination of focal length, the magnification would be positive and greater than unity. The nul point would therefore lie on the side of N1 remote from N2, and its position could be determined by the general method with greater or less accuracy than by the ordinary nodal-slide method according as it is at a distance from N1 less or greater than a respectively.
If the nodal points are determined by either of the two direct nodal-slide methods, which will here be distinguished as the first and second nul methods respectively, the principal foci and focal lengths may be most readily and directly found by measuring the distances from the nul point of the object in the first method and of the image in the second.
Now in obtaining the position of the image there will be a certain range along the axis within which the object may
lie and yet produce a distinct image on a fixed screen. And, conversely, there will be a certain range within which the screen may be moved and yet give a distinct image of an object in a given position.
The former range is called the depth of focus of the instrument, and may be shown (Heath's Optics,' pp. 269, 270) to be given by the expression
m tan a
where is the distance of the object from the entrance pupil, e the maximuin value of the circle of indistinctness, m the magnifying power, and a the angular aperture of the instrument.
In the second case the range through which the screen may be moved may be similarly shown to be given by the equation Δξ' =+
me tan a
(7) where' is the distance of the image from the exit pupil. In order to determine with precision the positions of the object and image the property required will be the inverse of these. In the first case, it will vary directly as the magnifying power and the angular aperture, and, in the second case, it will vary inversely as the magnifying power and directly as the angular aperture.
By comparing these magnitudes in the two cases just referred to it will be seen that the principal foci can be determined with greater accuracy by the first than by the second nul method.
Finally, the similarity of object and image will depend on the resolving power of the system, which increases with the angular aperture, and this again will be greater in the first nul method than in the second.
The conclusions arrived at in the foregoing were tested by the following experiments:
Two thin plano-convex lenses, each having a focal length of 25.8 cm., were mounted with their convex surfaces inwards and at a distance of 12.9 cm. apart. The calculated focal length of the combination was 172 cm., and the distances of the first and second nodal points were 8.6 cm. measured inwards from the first and second lenses respectively, so that the distance between the nodal points was 43 cm.
The positions of the nodal points were then determined by the first and second nul methods.
In the first method it was found that the lens system could be moved through a total distance 17 mm. before any sensible displacement of the image was caused by a small
rotation, whereas in the second method a motion through 5 mm. was required to produce the same result.
The principal foci of the combination were also found, a needle being used as object in the first method and to locate the image in the second method. The distances through which the needle could be moved before any sensible parallax was observed were 2 mm. and 4 mm. respectively.
This particular combination was chosen in order to get a fairly large distance between the nodal points, but as the lenses were uncorrected quite a small rotation caused the image to become confused owing to oblique aberration. A further test was therefore made with a 15 in. (38·1 cm.) Ross photographic lens in which the distance between the nodal points was only 1.6 cm. The following results were obtained for the ranges of adjustment in the two cases:
First Method. For nodal points 0-2 mm.; for principal foci 3.5 mm.
Second Method. For nodal points 1.5 mm.; for principal foci 11 mm.
These figures must be taken as indicating the relative rather than the absolute accuracies of the two methods, since the nodal-slide employed, though not of the roughest, had no fine adjustments. They appear, however, to show that the first nul method is in every respect superior to the second.
In conclusion, I wish to thank my colleague, Mr. J. E. Rycroft, for his kindness in making the diagrams.
IV. On the Value of the Mechanical Equivalent of Heat. By T. CARLTON SUTTON, B.Sc.*
HE following values of the Mechanical Equivalent of Heat (for references see Kaye and Laby's Tables' and Griffiths' Thermal Measurement of Energy') have been reduced to joules per mean calorie :—
The results obtained since the year 1905 are not as concordant as might have been expected. It may be of interest therefore to compare the value of the heat of vaporization of water at 100° C. obtained electrically at the Reichsanstalt by Henning (Ann. d. Phys. 1906-9) with that obtained by the author in terms of the mean calorie directly (Proc. Roy. Soc. April 1917).
Henning's results are shown in the following graph, which indicates that the value at 100° C. is 538.5 +0.3. Probably the constant errors are even less than 0.3.
Henning's work may also be checked by comparing it with that of Griffiths' at 30° C. The values for the heat of vaporization at that temperature are in good agreement, 579.9 and 579.3 mean calories respectively.
The corresponding values obtained by the author (Proc. Roy. Soc. April 1917) give 538.8±0.1 mean calories at 100° C.
If, then, Henning's mean value is correct to 1 part in 2000, the deduced value of the Mechanical Equivalent should be correct to within two units in the third decimal figure.
Henning uses the Jager and Steinwehr value of the 15° calorie to convert joules to calories, and takes the Reichsanstalt value of the e.m.f. of the Clark Cell. What he actually determines as the Energy of Vaporization at 100° C. is 538.5 x 4.188 Reichsanstalt joules.