PROBLEMS IN WATSON'S CO-ORDINATES. By THOMAS HILL, of Portland, Me.

[ABSTRACT.]

THE equation p—A (a—sin m )", in which, as in Vol. XXII, pp. 27-30, and Vol. XXIV, pp. 41-47, p is the length of a perpendicular from the origin on the tangent, and is the angle it makes with the axis, was discussed for the case in which n=-1. The curves represented by it are peculiar; and touch the conic sections in the parabola and circle, but not in any other forms; a new illustration of the fact that the capability of assuming certain forms in common is not proof of community of origin.

ON TWO NEW INSTRUMENTS FOR THE DETERMINATION OF TIME AND LATITUDE. BY S. C. CHANDLER, jr., of Boston, Mass.

I DESIRE to call the attention of practical astronomers to two new instruments for the determination of time, one of them also for latitude, the principles of construction of which I believe to be novel. They seem to possess advantages which entitle them to consideration.

The first of the instruments I propose to call the Almacantar, from an Arabic term, now obsolete, but appropriate for this purpose. From the results attained with an experimental instrument, I feel justified in claiming that in accuracy, efficiency, and convenience, this construction, for instruments of moderate size, is superior to the transit instrument, while it is very much cheaper. The principle involved is the same as that of Kater's floating collimator for determining the zenith point of a graduated circle. Beyond this, however, there is no further resemblance.

Briefly described, the instrument consists of a heavy base, with approximate levelling screws at the corners. From the centre arises an upright cylindrical pillar surmount by a cap of hard brass, and surrounded at the middle and the base of the pillar by

brass collars. These serve as bearings for a hollow brass sleeve, closely fitting and turning smoothly on the pillar. This sleeve is provided with a cross-head and lateral diagonal braces which support a shallow trough, in the form of a long, hollow rectangle. In this trough is contained the mercury, to the depth of about an

eighth of an inch, upon which swims a float of wood or iron, also a hollow rectangle, a little smaller than the trough. By means of two pins projecting from the sides of the trough and playing in vertical slots in the sides of the float, the latter is kept in place, while it is free to seek its equilibrium. From the middle of the inside edges of the float project two bent arms of brass, the lower ends of which support the horizontal axis of the telescope. The axis is provided at one end with a clamp, and at the other with an illuminating contrivance; and the telescope has a reticule of five horizontal spider-lines. (FIG. 1.)

If the telescope is turned on its axis and clamped at any desired altitude, and the whole instrument revolved around the upright axis, the sight line will describe a small circle parallel to the horizon. A few seconds only are required for the mercury and float to come to perfect rest. It is evident that the transit of stars, as they rise or fall over this horizontal circle, may be observed, and will furnish the means of finding the clock error, and the latitude, by a proper selection of the stars in different azimuths.

The mathematical theory of the instrument is succinctly indicated as follows: :

From the spherical triangle between the pole, the zenith, and

Let,,, and ., be assumed approximate values, which employed in eq. (1) give 0, and t ̧.

Also let ', ', be the apparent place of the star,

==

the actual zenith distance at which the instrument is pointed, r the refraction.

Then we have the final equation for the clock correction 4T, T being the observed clock time of the star's passing the middle thread,

4T=Z (z+r) + Ll + 0 −T + a + Dd

(5)

where (z+r), treated as a single unknown quantity, is the deviation of the collimation line of the telescope from the assumed horizontal small circle. The term LZ disappears if the correct latitude has been used in the computation of t ̧ and 0. The rest

of the second member contains only known quantities. For reducing the observed time T' of the star's passage over a side thread, whose interval (in time) from the middle or mean thread is f, we have

where the last term is insensible for observations made near the prime vertical, and is in general small except near the meridian.

For the special case where = 90

, that is, for the small horizontal circle passing through the pole, the above formulas are very much simplified, as follows:

In the practical use of the instrument at any given station, it will be convenient to compute once for all, for some one appropriate horizontal circle, or value of, the values of 0 and also the co-efficients D and Z for all Nautical Almanac stars, with