December 15, 1921] 



NATURE 



497 



how it got from tree to tree ; and in the resulting 

 discussion the fact was brought out that it had excep- 

 tionally tenacious slime. There appears to be no 

 doubt that it is carried by birds, to which it may 

 become attached at night while they are asleep. The 

 Island of Porto Santo possesses a large number of 

 endemic snails, and also a certain number of species, 

 such as Euparypha pisana and Cochlicella acuta (the 

 latter, new to the island, I found abundant locally 

 north of the Villa Baleira), which have been intro- 

 duced by man. But there is one species, Balea per- 

 versa, which was found by Wollaston only on the 

 isolated summit of the highest mountain in the island, 

 the Pico do Facho. This is far from human habita- 

 tion, and it is nearly certain that the snail was not 

 brought there bj- man. It is equally improbable that 

 it is a member of the ancient fauna, remaining un- 

 changed while all the other species have diverged in 

 various directions. The strong probabiliry is that it 1 

 was carried by birds, which could reach the Madeiras 

 in a short time, while the snails remained dormant 

 attached to their legs. Mr. J. Y. Johnson many 

 years ago cited no less than seventy species of birds 

 which had been obser\'ed as visitors or stragglers to 

 the Madeira Islands. 



The indications are, nevertheless, that few species 

 of snails can be carried long distances in this manner. 

 Balea is exceptional on account of its arboreal habitat 

 and tenacious mucus. There is, however, a small 

 species common in Porto Santo, the Heterostoma 

 patipercula, Lowe, which is also found in Madeira, all 

 three Desertas, and in the Azores. I have observed 

 that it adheres very tightly to rocks or other objects, 

 and is very tenacious of life. Specimens which I col- 

 lected in Porto Santo last January came to life and 

 crawled about the other day on being moved from 

 their positions. There can be little doubt, I believe, 

 that this little snail has been distributed by birds. 



It is even probable that certain species of Coc>;idae 

 -oale-insects) are dispersed in the same manner. On 

 the top of the Portello Pass, Madeira, I found a 

 strange little Coccid at roots of native grasses, and 

 expected it would prove undescribed. Mr. E. E. 

 Green, however, recognised it as Ortheziola Vejdov- 

 sfeyi. described from Central Europe. The larvae of 

 this insect could very well be carried on the feathers 

 of birds, and, escaping at almost any point on the 

 island, would be likely to find the necessarv- grass. 



T. D. A. COCKERELL. 



University of Colorado, Boulder, Colorado, 

 November 22. 



1^ The Cistributton of Brightness in the Penumbra during 

 an Eclipse of the ^^eon. 



On the day after the eclipse of October 16, Mr. 

 C. E. P. Brooks mentioned to me that the edge of 

 the umbra had seemed much more sharplv defined 

 when the eclipse was nearly total than at an 

 early stage. This obser\'ation suggested to me that it 

 might be worth while to work out the theoretical dis- 

 tribution of light over the lunar disc and gain some 

 information about the relation between umbra and 

 penumbra. The diagrams which commonly illustrate 

 the theory of umbra and penumbra in the text-books 

 are shaded in such a way as to suggest that there 

 is a sharp discontinuity in the illumination, the umbra 

 being indicated by one uniform shading, the penumbra 

 by another. 

 _ A little consideration shows however that the tran- 

 sition from bright to dark must be gradual. From 

 I the point of view of a spectator on the moon and 

 ^ within the umbra the sun would be entirely hidden 



NO. 2720, VOL. 108] 



by the earth. To a spectator in the penumbra but 

 close to the edge of the umbra the sun would be 

 nearly obscured, but to a spectator near the outer edge 

 of the penumbra almost the whole of the sun would 

 be visible. In traversing the penumbra the propor- 

 tion of the sun visible would increase slowly at first, 

 then more rapidly, the rate of increase would reach 

 a maximum with the sun about half obscured, and 

 finally the proportion would change but slowly when 

 the spectator neared the outer edge of the penumbra. 



The illumination received by various parts of the 

 moon is proportional to the area of the sun as seen 

 by the imaginary spectator, and if the local differences 

 in i-eflecting power could be ignored the apparent 

 brightness of the disc as seen from the earth would 

 follow the same rule. 



In Fig. I the small and large circles represent sun 

 and earth as seen from some point on the moon, 

 SP ( = r) is the apparent radius of the sun, EP ( = R) is 

 the apparent radius of the earth, and the angles PES, 

 PSE are denoted by P and jr— o respectively. The ratio 

 of I, the illumination of the moon at the point in 

 question, to I,, the illumination where there is no 

 eclipse, is determined by the fraction of the sun that 

 is not hidden, so that 



— = — 1(20-! 



In 2>rL\ 



^a)-^^[2^-sin2^J]. 



The angles a and 3 which occur in this formula 

 are related to x, the angular distance between the 



Fig. I. 



centres of sun and earth as seen from the moon, by 

 the equations 



R _ r^^ X 

 siua sin^ sin(tt — /3) 



For R and r the mean values i6-6o milliradians 

 and 464 milliradians may be adopted. On this basis 

 I /I, has been plotted as a function of x in Fig. 2. 



From this figure the values of x corresponding with 

 specified values of the ratio I /I, have been read off; 

 these are shown in the following table, the unit for x 

 being the milliradian : — 



I/I^ o 0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 x-o 



s 12-0 13-3 14-2 i4-<) 15-6 16-4 17-1 IT9 18-7 X9"6 ax-a 



(»-R+r)/ap o-oo 0-14 0-24 0-33 0-41 0-49 0-57 0-66 0-74 0-85 x-oj 



The angle x. which was defined as the distance 

 between the centres of sun and earth as seen from 

 a certain point on the moon, is to a ver>- close 

 approximation the same as the distance of the same 

 point of the moon from the centre of the shadow as 

 seen from the earth. 



