It was the consideration of these circumstances which led the Astronomer Royal to pronounce the transit of 1874 altogether unfit for the purpose of observing the durations of transit, as seen from opposite parts of the Earth's surface. And he suggested that four sets of observers should be sent to watch each of the four phenomena, accelerated and retarded ingress and egress; and that by determining the exact longitudes of their stations, and so (with the aid of exact chronometers) learning the exact Greenwich time of each phenomenon, the transit might be rendered available through the comparison of the results inter se. Clearly the elements of difficulty and the probability of error are seriously increased in this method as compared with one which practically requires but the simple estimate of duration, and scarcely admits of being affected by chronometer errors. However, let us note that by this method an observed difference of at the outside 24m. might be obtained;-not more, because the Sun cannot be observed when too close to the horizon, and because also of the difficulty of finding suitable stations.

Now let us see what can be done towards the utilisation of the transit of 1874 by the simpler method :

Suppose the northernmost station taken in latitude 60°, that is, along the uppermost parallel in the figure. As the whole duration of transit is but about four hours, and day lasts about six hours in this latitude on December 8, we may take a place two hours on the left of the central meridian in Plate XV., knowing that the same place will be (at the end of transit) two hours to the right of the central meridian in Plate XVI.; and at one epoch the Sun will be one hour risen, at the other one hour from setting. Doing this we find that the station (which lies in Siberia, not far from Lake Baikal) falls in Plate XV.) on the sixth cross-line from the centre, and in Plate XVI. above the tenth cross-line. In other words, the transit as seen from this spot exceeds the mean by (6+101), or 16 minutes.

Next for the southern station. Here we have a wide choice. If we put our observer on Petra Island (a place probably very little suited for astronomical observations) we get (from Plate XV.) ingress retarded by 8m., and (from Plate XVI.) egress accelerated by 12m., or in all the duration of transit falls short of the mean by 20m. If we take the place marked out by the Astronomer Royal for observing the transit of 1882, a place near Repulse Bay, in east longitude 105°, we get ingress retarded by 9m. and egress

accelerated by 9 m., or in all transit shortened by 181 m. If we take Victoria Land, in south latitude 70° (say), and east longitude 172°, we get ingress retarded by 6 m. and egress accelerated by 11 m., or in all transit shortened by 17 m. If we take Enderby Land, in east longitude 50°, we get ingress retarded by 11 m. and egress accelerated by 8 m., or in all transit shortened by 20 m. These four southern stations, combined with the northern station before considered, give a total difference of duration of 361 m., 344 m., 334 m., and 36 m. respectively. Also, as it would not be well to trust to a single northern station, it may be noticed that any part of the nearly circular region extending from Lake Baikal to Saghalien, and from north latitude 40° to north latitude 60°, might be used for observing the increased duration without important disadvantage as compared with the station already considered. Also, Crozet Island, Kerguelen's Land, and other parts of the Antarctic continent besides those considered, give abbreviated transits of considerable value. Thus for Crozet Island the abbreviation is no less than 17 in.; for Kerguelen's Land, 16 m. Even Macquarie Island, Royal Company Island, Hobart Town, and parts of New Zealand, might serve as useful subsidiary stations.

And now to compare the value of the transit of 1874 with that of 1882. We see that by the method of durations we get a difference of more than 36 m., whereas the maximum difference is 50 m. The Astronomer Royal has shown that for the transit of 1882 it is possible to take positions for observation (not by any means more favourable than those above considered) which give at the outside a difference of duration bearing to the maximum the proportion of 341 to 400. The maximum difference in the case of the transit of 1882 is only 32m. 48s., in place of 50m. 12s. as in 1874. Reducing 32m. 48s. in the proportion of 341 to 400, we obtain the period 27m. 57s. in place of the difference of 364m. which the most favourable situations in 1874 will give.

If we assume that the value of a transit is not to be estimated according to the magnitude of the observable difference, because the rate with which the planet crosses the Sun's limb is diminished in exactly the same proportion, and the error of observation correspondingly increased, we have the relative values of the transits of 1874 and 1882 as

36 to

501

28

321

or almost exactly as 6 to 7. But this extreme result, although as

R

it stands it is altogether opposed to the theory of the utter valuelessness of the transit of 1874, is obtained on an assumption which is unsupported by evidence. Mr. Stone has shown that the formation and breaking of the black ligament connecting Venus with the Sun at the true moment of internal contact is an instantaneous phenomenon in favourable weather. In unfavourable weather the error in the observation of this phenomenon should depend rather on atmospheric causes-the length of the periods of atmospheric disturbance, and so on-than on the rate of the planet's separation from the Sun's limb. If this is so, the transit of 1874 is superior to that of 1882 in the proportion of 36 to 28, or more than 9 to 7. If the truth lies between these extremes, the transit of 1874 may be fairly taken to have a value bearing to that of 1882 a proportion midway between 67 and 9:7; that is, the proportion of 15 14.

In any case no doubt can remain that the transit of 1874 is highly valuable, when dealt with in reference to the mode of observation we have been considering; and it seems clear that when all the difficulties and all the sources of error involved in the second method are duly considered, the simple method, founded on observed differences of duration, is to be held altogether more likely to give satisfactory results. I believe, therefore, that such preparations as geographers are already thinking of with reference to the choice of suitable southern stations for observing the transit of 1882 ought at once to be undertaken in connection with the transit of 1874.a

From the Quarterly Journal of Science for July 1869.

'Since the above was written the subject of the coming transits has been considered by Mr. Stone, than whom no one is better qualified to pronounce authoritatively on the principles which should guide us in utilising those phenomena. He is of opinion that observations made when the Sun has a less elevation than 10° would be altogether useless. This principle enables me to considerably augment my estimate of the relative superiority of the transit of 1874, with reference to the simpler mode of observation. In fact, the only southern stations which had seemed suitable in 1882 must at once be rejected; and thus we may say of that transit what had been said of the other, that the simpler mode fails totally' with respect to it. On the other hand, the value of the transit of 1874 is scarcely at all affected by the application of the principle.

Mr. Stone considers the superiority which I have ascribed to the simpler method to have a real existence, but to be so slight (the values of the two modes being as 6 to 5) as to be unimportant. This is just; but the distinction between this view and the imagined total failure of the method seems not the less to require attention.

CORRECTIONS OF THE ASTRONOMER ROYAL'S
STATEMENTS.

Ir is shown in the preceding article how the differences which exist between my results and those obtained by the Astronomer Royal, result from an adoption of a comparatively rough method of dealing with the problems involved. I now proceed to indicate the extent of these corrections, as also of those which have to be applied to the somewhat more exact work of the French astronomer Puiseux.

I first give a general summary of the changes under three heads :

1. The application of Delisle's method of absolute time differences. The relative as well as the absolute values of many stations are affected. Some which had hitherto appeared unsuitable are found to be unobjectionable. Others which seemed good appear unfit. In other cases the relative values of two stations are so affected that the results of a comparison between them are directly reversed. Lastly, many stations not hitherto thought of in connexion with the transit are found to be well suited for the application of Delisle's method.

2. The comparison between Delisle's and Halley's methods. Halley's method is found not merely to be applicable with advantage, which is all that can be said of it when central passages are considered, but to be superior to Delisle's,-slightly, when reference is made only to such stations as had been hitherto dealt with, noticeably when Antarctic stations are made use of.

3. The comparison between the Transits of 1874 and 1882 with reference to Halley's method. This comparison, conducted according to the principles laid down by Mr. Stone (than whom no one is better entitled to pronounce authoritatively on such points) shows that Halley's mode may be applied much more advantageously to the transit of 1874 than to that of 1882.

The results to be now brought into comparison for the sake of forming an estimate of the effect of phase, parallax, and the equation of time, upon the values of various stations, are these:

(1) Airy's, derived from the consideration of central passages, as supposed to be seen from the earth's centre, with the position-angles corresponding to external contact.

(2) Puiseux's, derived from the same phase, similarly seen, with the positionangles corresponding to central passage.

(3) Proctor's, derived from the consideration of internal contacts, as seen from the stations themselves, and with the position-angles corresponding to the phase so seen, and correction being made for the equation of time.

The following table exhibits the position-angles and epochs (for ingress) corresponding to these three orders of result:

In preparing the columns giving my own results in the accompanying tables, I have made use of the six orthographic maps forming Plates XIII., XIV., XV., and XVI. These were constructed with every precaution to ensure accuracy. The intersection of longitude-lines and latitude-parallels (to every 10°) were separately obtained by a double process of construction, and in all critical cases further tests were applied. In all, the construction of the maps involved upwards of 3,000 measurements. The results indicated in the maps have been also abundantly confirmed since the maps appeared, by the calculations and chartings made by others, and especially by the beautiful maps published by the American Government.

The six maps include four quarter-spheres, exhibiting the solar elevations and the coefficients of parallax. The other two exhibit the Earth as supposed to be seen from the Sun at ingress and egress (mean, and for internal contacts). In these the solar elevations are indicated by circles, and in place of 10 parallactic parallels, corresponding to the parallactic circles in the other map, there are laid down parallactic lines corresponding to minute intervals (the line across the Earth's centre being taken as a zero-line). These

The circles marked 9, 8, &c., pass through all the points at which ingress or egress is accelerated or retarded (as the case may be) by ths, ths, &c. of the maximum acceleration or retardation.