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APPENDIX A.

THE APPROACHING TRANSITS OF VENUS AND THE BEST MEANS FOR OBSERVING THEM.

Ox account of the interest attaching to the approaching transits, and more especially to the transit of 1874, for which a sum of more than 10,0007. has been voted by the Government, it seems desirable to exhibit here in a popular form the results to which I have been led by a very careful mathematical investigation of the conditions under which the coming transits may be most satisfactorily observed.*

Already in Chapter I. I have exhibited the general principles on which the determination of the Sun's distance by observation of transits of Venus depend. But as it is always advantageous in discussing astronomical relations to view them in as many aspects as possible, I now present a mode of

* I may remark here that I have gone over, at the expense of a considerable amount of time (for the work was altogether new to me), the process of calculation by which the elements of the transit are deduced from the tables of Venus and the Sun; but (as I anticipated when I began) the results I obtained accord so closely with Mr. Hind's, that the labour (save for the practice it gave me) was in a sense thrown away. It is worthy of notice, however, that M. Puiseux having obtained somewhat different results, this confirmation of Mr. Hind's results has a value which ordinarily would be wanting to researches of the sort. Mr. Plummer, of Mr. Bishop's Observatory, Twickenham, had already, at Mr. Hind's request, tested the published elements; and there can be now no doubt whatever that in this matter (as always) Mr. Hind's calculations are beyond question.

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considering transits of Venus which is perhaps the best and simplest conceivable.

Let s (fig. 92) be the Sun, v (Venus) and E (the Earth) travelling in the same direction and nearly in the same plane round the Sun as shown by the arrows, Venus the inore swiftly; and let it be noted that the dimensions of the Sun, Venus, and the Earth are here necessarily exaggerated, since even the Sun, if presented on the same scale as the distances, would be scarcely perceptible in our figure.

Now, Venus, like every other body in the solar system, throws a shadow, and the shadow is represented by the triangular space behind Venus in fig. 92. It comes to a point not very far (relatively) from Venus. Suppose this cone produced beyond its apex so as to form the shaded cone shown in the

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figure. It is obvious that to an eye placed within this shaded region Venus will be seen fully within the Sun's disc; and to an eye placed on the surface of this region Venus will be seen just touching the Sun's disc on the inside. Now conceive a double cone touching both Venus and the Sun, but having its apex between these bodies, as shown in the figure. Then obviously to any eye on the surface of the region between this apex and vv, Venus would be seen just touching the Sun's disc on the outside.* Clearly, then, to any eye between

*For instance, to an eye at v Venus would seem just to touch the Sun's disc on the outside at s, while to an eye placed at ', Venus would seem just to touch the Sun's disc on the outside at s'. This is obvious, because the lines v v, v' v, just touch the Sun and Venus, whereas any other lines from v and v to points on Venus pass clear of the Sun.

this surface and the region shaded in the figure, a part only of the disc of Venus will be seen on the disc of the Sun.

So that if an observer were carried through the double cone shown in fig. 92 behind Venus, he would see the following successive phenomena. When he came to the outer surface Venus would be in exterior contact, or as at a (fig. 14, p. 42); as he passed on to the inner surface Venus would enter more and more on the Sun's disc, until when he reached that surface she would be in interior contact, or as at в (fig. 14). Then as he travelled on through the inner cone Venus would seem to cross the Sun's disc, and she would just touch it on the inside when our observer reached the surface of this inner region on his passage outwards. Next, as he passed onwards to the surface of the outer region, Venus would be seen crossing the edge of the Sun's disc. And, lastly, as he passed that surface he would again see Venus in exterior contact, the transit thereupon coming to an end.

Now, during a transit of Venus the Earth does actually pass in such a way through these regions; or rather these regions overtake and pass over the Earth. In ordinary conjunctions the cones of fig. 92 pass above or beneath the Earth; but when the Earth is sufficiently near to the plane in which Venus is moving, the cones do not pass without encountering the Earth, and so a transit takes place. Further, the considerations in the preceding paragraph suffice to exhibit the general circumstances of a transit.

Since the cones overtake the Earth in the direction shown by the arrows, we may consider that the Earth passes through the cones in the contrary direction.

Suppose vv (fig. 93) to represent the same section of the outer cone as v v' in fig. 92; v v' the section of the inner cone; and E (fig. 93) the Earth, as shown at E in fig. 92. Then vv is really moving towards the left; but we are to suppose that E is moving towards the right through v v. Furthermore, if Venus is near an ascending node, as she will be during both the approaching transits, we must suppose the Earth to pass descendingly along such a course as E E' through

the region v v. The actual course, both as respects position and direction, is determined from the calculated elements of the transit. With this calculation we need not here concern ourselves. The figure shows the course actually traversed by the Earth in 1874 and 1882.

Now, taking the Earth through vv for the 1874 transit, let us consider the various critical points, so to speak, of her When she first touches the outer circle v v' external

course.

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contact will have begun at that point of the Earth which first reaches this circle. She passes on, falling more and more within v v', until she is just wholly within. All this time external contact is taking place wherever the outline v v' intersects the Earth's disc; at parts within that line Venus is seen partly

* As to the size of v v' compared with that of the Earth, it is easily seen from fig. 92 that v v' is less than a great circle of the Sun, very nearly in the proportion that the Earth's distance from Venus exceeds the Sun's. If the cone of which vv is a section had its vertex at Venus's centre this proportion would be exact.

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