Page:Philosophical magazine 23 series 4.djvu/40

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34 The Astronomer Royal an the Direction of the Joints

axis, depends upon the velocity of light whose vibrations are parallel to that axis, or whose plane of polarization is perpendicular to that axis.

In a uniaxal crystal, the axial value of E will depend on the velocity of the extraordinary ray, and the equatorial value will depend on that of the ordinary ray.

In "positive" crystals, the axial value of E will be the least and in negative the greatest.

The value of $D_{1}$ , which varies inversely as $E^{2}$ , will, cæteris paribus, be greatest for the axial direction in positive crystals, and for the equatorial direction in negative crystals, such as Iceland spar. If a spherical portion of a crystal, radius = $a$ , be suspended in a field of electric force which would act on unit of electricity with force =I, and if $D_{1}$ and $D_{2}$ be the coefficients of dielectric induction along the two axes in the plane of rotation, then if $\theta$ be the inclination of the axis to the electric force, the moment tending to turn the sphere will be

{\frac {3}{2}}{\frac {\left(D_{1}-D_{2}\right)}{\left(2D_{1}+1\right)\left(2D_{2}+1\right)}}I^{2}a^{3}\sin 2\theta ,

(143)

and the axis of greatest dielectric induction ( $D_{1}$ ) will tend to become parallel to the lines of electric force.

IV. On the Direction of the Joints in the Faces of Oblique Arches.

By G. B. Airy, Esq., Astronomer Royal^[1],

MY attention was lately called to the following passage in Mr. Buck's 'Essay on Oblique Bridges,' 2nd edition, p. 7. "After having had several drawings of the faces of oblique arches made on a large scale and projected with great exactitude, we observed that the following remarkable property exists. If the lines, which are the chords of the small curves forming the joints in the face of the arch, be produced, they will all meet in one point O, below the axis of the cylinder; and this property was found to hold even when the obliquity is so great as to depress the point O out of the cylinder altogether." The author then determines the point O by geometrical calculations for the joints at the spring of the arch, and, as far as I can perceive, makes use of this empirical theorem for determining the directions of all the other face-joints.

The theorem is perfectly correct; and the discovery of it bears testimony to the accuracy with which the author's plans must have been drawn, in a process of rather difficult geometry, and to the care with which they have been examined. The theorem^ moreover, is true in the utmost generality, as regards the extent

↑ Communicated by the Author.

[1] Communicated by the Author.

[1]