June 17, 1909] 



NATURE 



46; 



THE PROBLEM OF AN ULTRA-NEPTUNIAN 

 PLANET. 



IN this memoir ' Prof. W. H. Pickering- first gives 

 a graphical method by which Neptune might 

 have been discovered from the data used by Le Verrier 

 and Adams. Having thus demonstrated the practical 

 value of his method, he proceeds to apply it to a 

 search for a planet beyond Neptune. He calls his 

 hypothetical planet by the letter " O," obviously as 

 the ne.xt letter of the alphabet to the initial of Neptune, 

 and he finds, inter alia, 



Mean distance 51 '9 



Period 373'5 years 



Mass... ... ... = twice that of the earth 



R.A. in" 19090 7h. 47m. 



Now the problem presented by Uranus, Neptune, 

 and " O " may be very readily reduced to the known 

 problem, already fully worked out, of Mercury, Venus, 

 and the earth, for it is easy to show that the theory 

 of a pair of planets is the same if we retain the masses 

 but alter the distances of both in the same propor- 

 tion. 



First of all, as, roughly speaking, we are only able 

 to observe heliocentric longitudes of Uranus and 

 Neptune, we must suppose that our fictitious observer, 

 to whom the existence of the earth is unknown, is 

 only able to observe heliocentric longitudes of Mercury 

 ttnd \'enus. We may speak of him, therefore, as an 

 observer in the sun. 



Secondly, as the mass of " O " is twice that of 

 the earth, we must credit the observer in the sun with 

 instruments of twice the precision of those used by 

 ourselves. 



Lastly, we may divide all distances by 51, and all 

 time intervals by 365. This latter factor enables us 

 to substitute days for years. 



We are. therefore, to suppose that an observer in 

 the sun, with instruments of twice the accuracy of our 

 own, has observed Mercury for four months and 

 A'enus for two months, and that in addition he has 

 one or two stray observations of Mercury and Venus 

 made before he recognised its planetary character. 



Now let us turn to the tables of Mercury and Venus, 

 •and estimate for ourselves what chance such an 

 •observer has of demonstrating the existence of the 

 earth. 



Before doing this, however, we must pause for a 

 moment in order to show that we may dismiss from 

 notice all long-period terms. These terms play a very 

 conspicuous part in planetary theory. Their existence 

 depends upon the same principle as that of the swing, 

 where a very small force applied at regular and suit- 

 able intervals will produce very large oscillations. 

 Their existence necessitates the expansion of the dis- 

 turbing function to ten or a hundred times the 

 accuracy otherwise necessary, and the consequence is 

 that from the computer's point of view the short- 

 period terms are dismissed with scanty notice in the 

 account that he gives of his work. 



For example, in the heliocentric longitude of Venus 

 there is a term 



3"sin(i3E-8V) 



with a period of 239 years. Our hypothetical observer, 

 in the sun would have no chance of detecting such a 

 term as this. If he detected in two months' observa- 

 tions any term at all it would be one of the follow- 

 ing : — 



5" sin (V - E) 



n"sin 2(V-E) 



rsin3(V-E) 



•lie periods being 5.S5, 292, and 195 days respectively. 



1 '' .\nnals of the .-Vstronomical Observatory of Harvard College." Vol. 

 Ixi., part ii. A Search for a Planet beyond Neptune. By W, H. Pickering. 

 ((Cambridge, Mass. : The Observatory, igog.) 



NO. 2068, VOL, 80] 



It may further be observed that it is the long-period 

 terms which are largely affected by small changes in 

 the elements of the disturbing planet; the short- 

 period terms are not appreciably affected. During the 

 hypothetical two months of observation the question 

 is, " Where is the disturbing planet at that time? " 

 not " Are the elements such as produce a long-period 

 term? " 



Further, it is not sufficient that a term of moderate 

 amplitude and period should exist. In the hypothetical 

 two months, only a ver)' small portion of a complete 

 period is observed, and the conditions must be such 

 that the term is not mistaken for uniform motion 

 or for a term periodic in the period of the planet 

 under observation, for in either case the term could 

 be represented by a change of the elliptic elements of 

 the orbit. 



-At this point we wish to say that we do not think 

 Prof. Pickering's case is a good one, and, having thus 

 proclaimed ourselves adverse critics, we wish imme- 

 diately to concede the following point. In our opinion 

 the hypothetical observer could detect from two 

 month's' observations the term 7"sin3(V— E) in the 

 heliocentric longitude of Venus, and could distinguish 

 it from a mere error of assumed elliptic elements, 

 provided only the phase of the argument happened to 

 be suitable during the period of observation. Revert- 

 ing now to the actual case, the existence of planet 

 " O " could be demonstrated from the observations of 

 Neptune at the present date if the epoch of planet 

 " O " were suitable. If the observations of Neptune 

 show nothing-, the hypothesis of the existence of " O " 

 would not be negatived, but the hypothetical " O " 

 would at least be confined to certain limits of longi- 

 tude. 



Prof. Pickering, however, has based his discovery 

 of " O," not on Neptune, but on Uranus. In the 

 hypothetical case we have to consider the perturbations 

 of Mercury by the earth. In this case we have terms 

 such as 



0-2" sin (M-E) 

 0-3" sin 2(M-E), 



which we believe to be quite incapable of detection 

 from four months' observations, or at any rate of 

 detection and distinction from elliptic terms. In 

 saying this we do not forget that the terms must 

 be doubled to take account of the double mass attri- 

 buted to " O." 



It will be seen that our criticisms are directed 

 against Prof. Pickering's figures as we find them. We 

 have argued against the extremely small mass 

 assigned to the hypothetical planet, seeing that Prof. 

 Pickering's data are the observations of Uranus. 

 While constructing our argument we have, however, 

 convinced ourselves that the time is ripe for a discus- 

 sion of the observations of Neptune, for if the planet 

 " O " exists, or anv appro-ximation to it, it should 

 have produced, or at any rate should soon produce, a 

 visible effect on Neptune. 



NOTES. 



Meetings of two special commissions appointed by the 

 International Meteorological Committee at Paris in 1907 

 will be held in London during the week commencing- 

 Monday next, June 21. The appointment of the first com- 

 mission arose out of a proposal made at Innsbruck by the 

 Rev. Lowis Froc, S.J., -director of the Zi-ka-wei Observa- 

 tory, for the general adoption of a code of maritime 

 weather signals now in use in far eastern waters, and a 

 further proposal made at Paris by Prof. Willis Moore, 

 chief of the United States Weather Bureau, in favour of 

 an international system of maritime weather signals. To 



