1874 be called for; since I have been able to demonstrate that without seeking such dismal regions, a more favourable opportunity will be afforded in 1874 than in 1882, even on the supposition that the selected stations were visited during the latter transit. And this result has been confirmed by mathematicians of eminence.

As regards other southern stations in 1882, it needs but a brief study of figs. 95 and 96, to see that none are available. All the stations on the right of the arrow-line in fig. 95, ard close by the arrow-feather, are unseen in fig. 96—that is, the Sun has set at those stations. The barren Antarctic land south of Cape Horn (Graham Land) is not very favourably situated in fig. 95, but in fig. 97 has passed over to the wrong side of the central cross-line; in other words, there is acceleration of egress instead of the needed retardation. Similar remarks apply to the southern parts of South America.*

So far, then, is it from being true that Halley's method is wholly inapplicable in 1874 but is applicable under favourable circumstances in 1882, that the exact reverse is the case. Halley's method is even better than Delisle's in 1874, and is wholly inapplicable in 1882.†

The reasoning in the note beginning on p. 37 will be found to be well illustrated by a comparison of Plates IX. and X. with figs. 95 and 96. It will be seen that in the transit of 1874 the rotation of the Earth carries the northern stations the wrong way as regards that lengthening which is required; while in the transit of 1882 the northern stations are carried the right way for the shortening which is required. As to the southern stations, it will be seen how, according as they lie above or below the pole (as shown in the maps), the lengthening or shortening can be increased or diminished. It is the fact that no stations below the pole are suitable in 1882, which spoils the otherwise sound reasoning of the Astronomer Royal.

+ It will be seen from fig. 93 that the path of the Earth crosses the surface of the shadow-cone much more obliquely in 1874 than in 1882; and, as this path is as it were the reflexion of the apparent path of Venus over the Sun's disc, Venus crosses the limb much more obliquely in 1874 than in 1882. Thus the actual progress of ingress and egress must take place much more slowly. Mr. Stone believes that all time-errors will be correspondingly increased. The comparison above referred to is founded

G G

And now I propose to indicate, in conclusion, considerations which lead me to believe that the direct method of determining the actual parallactic displacement of Venus on the Sun's disc can be applied under very favourable circumstances during the approaching transits.

Let us first consider if we cannot in part diminish the difficulties which seem inherent in this method.

Suppose an observer at one station sees Venus at a given epoch as at a (fig. 97, I.), while at the same epoch another observer sees her at B. Then it is perfectly obvious that in such a case as is illustrated by I., when the two observations

Illustrating the displacement of Venus as seen from different
terrestrial stations during transit.

are brought into comparison, any error affecting the bearing of B or A from C will tend to produce its full effect in increasing or diminishing the parallactic displacement separating A and B. But now suppose that a and b in II. represent the

on this opinion, very unfavourable to my views. I must, however, express my own conviction that though the time-errors will be larger in 1874 they will not be proportionately increased. The slowness of ingress and egress will give the observer an opportunity of observing the features of ingress and egress more satisfactorily. Especially if my suggestion is attended to as regards the breadth of the ligament at the moment of breaking or forming (see p. 63) will this be the case. Apart from this, as all the time-differences are increased precisely in proportion to the slowness of ingress or egress, a very slanting passage is cæteris paribus as favourable as a more direct one.

true apparent positions of Venus as seen from the two stations -a, b, and c (the centre of the Sun's disc), lying on, or nearly on, a line. Then it is clear that a small error of bearing will produce no appreciable effect in increasing or diminishing the parallactic displacement separating a and b.

Hence, since distances are much more easily determined than bearings in observations such as are here considered, it is obvious that if we select pairs of stations where the parallactic displacement of Venus will be along a radius of the Sun's disc, the resulting estimate of the distance of Venus is much more likely to be correct than in any other possible

case.

Now, if we inquire what are the conditions to be fulfilled for this purpose, we shall find it exceedingly easy to obtain a geometrical means of determining what the answer should be. Lines drawn to Venus from two stations on the Earth meet the Sun's disc on a radius, if they are to fulfil the required conditions. In other words, these lines are in a plane which passes through the Sun's centre, and of course they are in a plane through Venus's centre, since each passes through that centre. Hence they are in a plane through the axis of the cones shown in fig. 92; for this axis passes through the centres of Venus and the Sun. But a plane through the axis of these cones must intersect such a section as v v' in figs. 92 and 93 in a radial line. Hence the two terrestrial stations must at the moment of observation be on a line through o in fig. 93. Now, the slanting lines over Plate VIII. are such radial lines; and therefore any two stations which in any figure of this plate lie along one of the slanting lines fulfil the required conditions. They must, however, be as far apart as possible. For example, taking the fifth picture of the Earth in Plate VIII., we see that Japanese stations and a station near Crozet Island could be very effectively combined at the hour named under this particular presentation of the Earth.

Of course the best of all such opportunities would arise when the two stations lay near opposite extremities of one of the cross-lines traversing the middle of the Earth's disc.

The various stations which fulfil the required conditions can be very readily determined from a study of Plate VIII.

The application of photography, as proposed by Mr. De La Rue and Lieut.-Col. Tennant, would be greatly facilitated by the choice of stations on the principle here enunciated. (See, further, my paper in the Monthly Notices of the Royal Astronomical Society,' No. 3, vol. xxx.)

The following tables, which I have calculated with great care, supplement this essay, and serve to exhibit the qualities of the principal stations suited for observing the transit of 1874, either as respects De Lisle's or Halley's method :