- r,

and one point of the slit, we see that they will strike the lens at an angle of incidence about equal to i, will traverse it in a plane which we will call the plane of refraction inclined to the first plane i and finally emerge in a plane parallel to the first. The plane of refraction will intersect the lens along two circles whose distance apart at the centre will be greater than the thickness of the lens in the ratio of cos r to 1; hence their radii R' will be less than the radius of curvature R of the surfaces of the glass in the same ratio, or R' — R cos r. Again, the apparent index of refraction n' will be different, and 1 2 (n'-1) and =

since

}

=f cos r

=

2(n-1)
R

n 1
"'-1'

=

R

=

R

n.

we have f':

Fig. 2.

B

C

It therefore only remains to determine n', the apparent index of refraction. As the problem is one in spherical trigonometry, suppose a sphere described around the centre of the lens and projected in Fig. 2, the eye lying in the axis of the lens prolonged. Let CA, the angle through which the lens has been turned, and CE r, the corresponding angle of refraction. Then if the surface of the glass is vertical, as at the centre of the lens, the incident ray will be AC and the refracted ray CE. Next suppose the surface slightly inclined by the amount CD BC=v, as is the case for the upper and lower parts of the lens. AB i will now be the angle of incidence; and, to construct the refracted ray, we have first the condition that sin i' — n sin r', and secondly that the incident and refracted ray shall lie in the same plane with the normal BCD. To construct it, pass a plane through the normal BC and the incident ray AC, which will intersect the sphere along the great circle AB and FD; on this, lay off DE =r' such that n sin sini, but, as v is infinitesimal, will be sensibly equal to i, and r' to r. Now in the right-angled spherical triangle FCD, sin CD=sin DF sin CFD, or sin v=v=sin i × F, or F= ;; and in the triangle FEE', sin EE' = sin FE' sin EFE', or

v

=

sin (ir) F, or, substituting the value of F just found, EE' sin (i-r) Calling and the angles of incidence and refraction of the ray with regard to the section of the lens made by the plane of refraction, then i" will not equal BC, but will be the angle which, when projected on the plane of the section of the lens, will

gives f'=f

which, with the above values, gives n'

sin i cos r

stituting this value in the equation ƒ'=ƒ cos r
sin i sin (i − r) cos r
sin (i-r)

(n − 1)

(n-1)

(n' -1) This would be the focal distance if the rays on emerging remained in the plane of the section of the lens. But they pass into a plane inclined to this i r, hence the observed focus f" will be such that when projected on the plane of the section it will equal f', or f" cos (ir)=f'. Hence finally f"=ƒ

sin i sin (i―r) cos r
sin (ir) cos (i —r) *

(n-1) This last step may be open to criticism, but the close agreement with observation seems to justify it. In Table IV., this formula is compared with observation, as the law for the vertical slit is compared in Table III. The columns in the two tables correspond, and it will be noticed that the agreement is very close.

The principal practical application of these results is to photographic lenses. It will be seen that a single lens, even if perfectly corrected for spherical and chromatic aberration, is still subject to this defect. Con

structing the curves with polar co-ordinates, taking the radius vector equal to the focal length and its angle equal to the angle of incidence, we obtain a line every point of which would be in focus at the same time. This shows that in a photographic camera for lines passing through the axis, corresponding to the vertical slit, the surface instead of being a plane should have a radius of curvature of only .3 the focus. For lines perpendicular to these, or circles concentric with the centre, corresponding to the horizontal slit, the curvature should be .7 the focus. We also see the importance of having telescope lenses carefully centred, and why the images of stars, if this adjustment is not exact, are elliptical instead of circular.

Since writing the above, a further application of these formulas has been suggested in the case of the eye, that the imperfect vision at a distance from the centre of vision may be due to the rays passing obliquely through the lens. It will also be noticed that the curvature of the retina corresponds nearly with that which would give the best vision. As stated above, for radial lines the radius of curvature should be about .3, and for concentric circles .7, its distance from the lens. The actual curvature in the normal eye is about .5, or the mean of these values.

VIII.

BRIEF CONTRIBUTIONS FROM THE PHYSICAL LABORATORY OF HARVARD COLLEGE.

No. I.

ON THE EFFECT OF HEAT UPON THE MAGNETIC SUSCEPTIBILITY OF SOFT IRON.

Br H. AMORY AND F. MINOT.

Presented, Jan. 12, 1875.

THE determination of the question whether heat influences the capabilities of soft iron to be magnetized appears to us to be an interesting question, since, in the later forms of magneto-electric engines, the armatures necessarily become heated by their movement in a magnetic field. The question is also of interest from a molecular point of view. We have confined ourselves to the determination of the effect of such heating upon the induced currents produced by suddenly passing an electric current about the bar of soft iron, which is heated to different temperatures.

Great difficulty was anticipated at first in determining the temperatures of the bar at different times. Preliminary experiments show, however, that the question resolved itself into observing the decided changes at the temperature of dull red heat and at white heat. The first method of experimenting was as follows: bars of soft iron, 1 cm. in diameter, were placed so as to form the armatures of the electromagnets; a coil of fine wire, the induction coil, was slipped upon these bars, forming the armatures, and the curve was drawn, which showed the distribution of magnetism over the armature when the electromagnets were excited; then the bar was heated, and the change in the curve noted. The induction coil was so placed that its plane was at right angles to that of the coils of the exciting electro-magnets. This apparatus showed a slight increase of magnetic susceptibility in the bars of soft iron as they were heated. The magnetic state increased up to the point when the bar began to change slightly in color from the effect of the heat; it then remained constant. Owing to the diffi

culty of heating bars of comparatively large diameter to a point beyond that of dull red heat, this method was abandoned, and the following was adopted. The testing apparatus consisted of an electro-magnet, horseshoe in form, but the wire of which was placed at the bend of the horseshoe, so that the electro-magnet was practically a straight one, with a horseshoe-shaped core. Upon one of the limbs of the horseshoe the induction coil was slipped, so as to still remain at right angles with the electro-magnet. The soft iron bar or wire was then made the armature of the electro-magnet. It was found that this arrangement was a very sensitive one; for any change in the condition of the wire forming the armature was immediately shown when the electro-magnet was excited, and an induced current passed through the fine induction coil. This method allowed us to experiment with wires, or bars of any suitable diameter. For, as it will be shown later, the size of the armature had very little effect upon the strength of the induced currents produced at making and breaking the current in the electro-magnet. We shall speak of the horseshoe-shaped core and the armature as a magnetic circuit, which of course is a mere convenient term. When the armature is applied to the poles of the horseshoe, and the electro-magnet is excited, then such a circuit may be said to be closed. The bars or wires were tested at a dull red heat, and also at white heat. The first bar used was 2 mm. in diameter, and the following table shows the Six observations were taken, at intervals of one

results obtained.

minute apart.

When the magnetic circuit was closed by the armature, the first induced current which was produced by making the circuit of the electro-magnet was greater than the succeeding ones. This was doubtless due to residual magnetism. We do not speak of the induced current which resulted from breaking the circuit of the electro-magnet; for this was equal to that produced by making the circuit. The mean of the two renderings gave the correct result.