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and this value substituted for E in the second equation gives
(m—m')3 &2 + (l + nt)2 = (P—P):x,
a formula similar to that which was found for eclipses of the Moon; except that d2 is multiplied by
in this, and not in the other. But, as the
same factor is also found in an inverted order in the values that are to be substituted for d, the results will be absolutely the same for both. This equation is therefore to be resolved by adopting the same auxiliary angle, and eliminating m-m in exactly the same manner as has been done in the preceding article, and we shall obtain
t= (+ sin a
́(P—P'′)'d3—l' cos2a—l sin3a).
All that is now required to be done in order to find the epoch answering to any particular phase of an eclipse is to substitute, in this expression, the value of d answering to that phase; and this substitution will give the corresponding value of t, or the time of its
For example, if the beginning and end of the eclipse were required, or the moment when the edge of the Earth's disc entered and quitted the lunar penumbra, it would only be necessary to substitute for d its value at that time, viz.
D + D'
in which the letters denote the same quantities as before; and the quantity under the radical corresponding to these two instants would become
the other quantities remaining as in the preceding equation. The same conclusions may also be derived in this case relative to the middle of the eclipse, the extent of the part eclipsed, and the duration of the eclipse, as in the preceding article. If the observer were supposed to be situated at the centre of the Earth, p=0, and p'0 also, and therefore the part under the radical would then be simply
We shall subjoin an example for the use of such of our readers as may wish to try the calculation; and as it is designed only as an example of the calculation, we shall select the eclipse of the Sun, which happened on the 1st of April, 1764, and was adopted by Dionis-du-Sejour, as the basis of his great work on eclipses, taking the elements as they stand in that work; the only difference between this example and others being that of finding the elements by means of astronomical tables for the proper time.
h. m. S
Time of the conjunction in solar time at Paris = 10 31 5
Moon's apparent diameter
By substituting these quantities in the formula, we
a=5° 44′ 27".
At the commencement of the eclipse t=
At the middle
And, at the end
+ 29 39
+ 2 27.7
54 1.5 29 29
--- 0 8
= 2 35 59
The beginning and middle of the eclipse, therefore, happened before the conjunction, and the end after it. The least apparent distance of the centre of the Earth and the lunar shadow, which was equal to
Р -l cos a=39′ 27". This distance being less
than the parallax of the Moon, which is equal to the apparent semidiameter of the Earth seen from the Moon, the shadow, at that time, fell wholly upon the terrestrial disc, and, consequently, the eclipse was central in some places. Since the apparent diameter of the Sun exceeded that of the Moon, the eclipse was therefore central and annular in certain places; and, in others, it was annular without being central. There also remained more than half the terrestrial disc where it would not be seen at all; for the radius of the penumbra 30′ 45", which is less than the least distance of the centres 39′ 27′′; so that on the radius over which the shadow passed there remained a part 8′ 42′′ not eclipsed. Subtracting this from the apparent semidiameter of the Earth, equal to 54' 1".5, there remained 45′ 19′′.5 for the part eclipsed; and which is easily found to be 5.03 digits.
With respect to the slight modifications necessary to be introduced into the results obtained by the formula we have given, when the greatest accuracy is required, both on account of the variations in all the elements of the Moon experienced during the eclipse, the figure of the Earth, the effect of its atmosphere, and the consideration of a given physical point on the Earth's surface, the nature of this work obliges us to refer the reader to some comprehensive treatise on astronomy. He will find them treated in detail in the Traité élémentaire d'Astronomie Physique, par M. Biot; whose method we have chiefly followed in our present disquisitions on this subject.
The Naturalist's Diary.
Hail, eldest of the monthly train,
As Winter unfolds his awful train, vapours, and clouds, and storms,' the contemplative observer of nature becomes habituated to views of the stupendous and sublime. Verdant groves, variegated meadows, and radiant skies, are now succeeded by leafless woods, dejected wastes, and a frowning atmosphere. But while the incurious and inattentive perceive a dreary uniformity in all around, the penetrating eye of the rural student discovers many a varied aspect of beauty and excellency, which still invite to the most pleasing investigation. And, however paradoxical it may appear, he finds inexhaustible sources of serenity and delight, in that mood of melancholy musing on scenes of desolation, which, in vulgar estimation, would rather
Deepen the murmur of the falling floods,
And breathe a browner horror o'er the woods. In fine, in each vicissitude of the seasons, he still discerns the omnipotent Creator, ever bountiful to man; and, whether the gentle gales breathe propitious in spring, or resistless storms ravage the earth in winter, his cultivated mind kindles with devotion, and even calls upon the inanimate world to join him in adoration :
TO HIM, ye vocal gales,
Breathe soft, whose spirit in your freshness breathes:
Where, o'er the rock, the scarcely waving pine
Rain and wind are extremely prevalent in this month, and, as the frost seldom sets in till the latter end of December, this month may be reckoned the most unpleasant of the whole year. The weather is cold, bleak, and gloomy; and, generally, one continual succession of storms and tempests. At other times, we have ice and snow; and, about Christmas, the cold is sometimes intense.
At WINTER'S numbing touch, the fields
The lifeless leaves blow here and there,
And here and there the wood-birds flit,
In the unfinished furrow lies
The plough, nor wounds the field;
Shorn of his rays, scarce does the sun
But sudden sets :-Night fast behind
The flowers mentioned as continuing in blow in January, of course afford their beauties in this month. Evergreens, firs, ivy, laurel, and that most beautiful plant the arbutus, rich in flowers and fruit at the same time, serve to enliven this dreary month. The common arbutus (arbutus unědo), or strawberry-tree, rises to the height of twenty or thirty feet, but rarely with an upright stem: it usually puts out branches very near the ground. The leaves keep on all the winter, and are thrust off in the spring by new ones; so that it is always clothed with leaves.
The oak, the beech, and the hornbeam, in part, retain their leaves, and the ash its keys. The common holly (ilex aquifolium), with its scarlet berries, is now conspicuous; and those dwarfs of the vege