Page:Proceedings of the Royal Society of London Vol 1.djvu/463

Rh number of angles P S Q, Q. S Q. be also taken, each equal to M S I’, the angle which the ﬁrst drawn radius makes with the axis, then will the continued product of all the radii S P be equal to the last S Q multiplied by the latus rectum raised to the power of n -— 1, n being the number of angles taken.

The author thence proceeds to deduce other theorems that would be for the most part complicated and unintelligible when geometri- cally enunciated, though suﬂiciently simple in their algebraic expres- sions. They are indeed, as the author observes, properties rather of the equations of the conic sections, than of the curves themselves; properties of a limited number of disjoined points, determined ac- cording to a certain law, rather than of a series of consecutive points composing a line.

In the course of this investigation the author employs one species of notation, which is new, and for which he apologizes, by explaining its advantage in point of simplicity.

Since a minute description of the new circular instrument, which has been lately put up at Greenwich, is intended to be given to the Society as soon as it is completed in every respect, the Astronomer Royal takes do further notice of its construction than is necessary to show by what means the results of his observations of the sun at the last solstice was obtained.

In other instruments, which take their point of departure from a. plumb-line or level, the zenith distance of the sun is the primary ob- ject of investigation; and the polar distance of the sun, which is the ultimate object, is obtained by adding the co-latitude of the place, which completes the entire are.

But by the mural circle at Greenwich, to which there is neither level nor plumb—line, the total are may be measured without any exact knowledge of the zenith point; and the co-latitude, which in all other cases it is so essential to know correctly, becomes an object of mere curiosity, rather than of real necessity.

It is, however, convenient to assume some imaginary point near the zenith, the position of which, with respect to the ﬁxed stars, may be determined within one tenth of a second; and from this imaginary point Mr. Pond measures the distances of the sun southward, and of the pole northward, as the best means of obtaining the entire arc; but he also adds a computation of the same solstitial place of the sun, as obtained by direct measurement from the pole without the aid of his imaginary intermediate point, and the difference is found to be only 0'15 of a second.

In the determination of this arc, it is evident that, however accurately it may have been mechanically determined, it must still be