54 



NATURE 



[May i8, 1893 



•during the exciting conditions of an eclipse, but may also have 

 arisen from the fact that the actual eclipse wa, shorter than the 

 •calculated one. T. E. Thorpe. 



Daylight Meteor, March i8. 



This meteor, reported to Nature by Dr. Rorie of Dundee, 

 was also seen by Mr. A. G. Linney at Ackworth, near Ponte- 

 fract. Careful comparison of their records gave a probable 

 path from just S.E. of Lanark to 30 or 40 miles W. of Mull. 

 Notes received later from the Fort William Low Level 

 Observatory make it probable that the end ■was nearer there, 

 say just N. of Mull. The foraier gives an actual path of 180 

 miles, from a height of 140 to 42 miles ; the latter, 140 miles, 

 ending at a height of 40 miles or less. If Dr. Rorie's time is 

 correct, it travelled at a rate of 36 ox 2% miles per second, both 

 being rapid. This accounts for the magnificent streak. As 

 this floated across to Dundee in three quarters of an hour, the 

 central part must have in that time travelled 95 or 85 miles at a 

 height of 100 to 90 miles above the earth, and in an E.N.E. 

 direction. Thus its velocity seems to have exceeded 100 miles 

 an hour. The KrakataTj dust reached us in the same direction, 

 its greatest height being 30 to 40 miles, and speed 72 miles per 

 hour. A greater speed at greater altitude quite agrees with 

 theoretical probabilities, although the increase seems very 

 great. J. Edmund Clark. 



Roche's Limit. 



A LETTER has been addressed by Mr. D. D. Heath to the 

 Editor of Nature on the statical problem involved in 

 "G. R.'s" approximate method of finding Roche's limit. 

 This letter has been submitted to me, and I have thus 

 been led to look more closely into "G. R.'s" proof, which 

 I adopted ina recent letter to Nature (April 20, 1893, p. 581). 

 Mr. Heath shows that both " G. R." and I have omitted the 

 factor 2 from our result, and I now see besides that a statical 

 soluiion is insufficient for the problem in question. 



The problem may be stated thus ;— To find at what distance 

 two equal spheres in contact can revolve in a circular orbit 

 round a third, the centres of the three spheres being in a straight 

 line. 



Take the following notation :— The single sphere of density 

 <r and unit radius ; the two spheres each of unit density and 

 radii r ; c, the distance from the centre of the single sphere to 

 the point of contact of the two ; and la the angular velocity of 

 the system. 



The problem may be rendered statical by introducing the 

 conception of centrifugal force estimated from the centre of 

 inertia of the system, which is also the centre of rotation. The 

 distance of the centre of inertia from the point of contact of the 

 two spheres is cal{(r + 2r^). 



Then the three equations, only two of which are independent, 

 which express the equilibrium of the spheres are : — 



\<r + 2r^ I {c - rf (2/-)-' 



V <r + 2r'f 3 \(c + rj^ {c - r)-) 



Adding the first two of these and dividing by ?ir<r, and then 



3 

 subtracting the second from the first and dividing by ?ir?-, we 

 Iiave 



>Y f_ \ ^ 4(^' + »-') 



» \<r + 2;^/ {c" 



3^_ J Sc<r 



Eliminating «' we have 



C(C^ - ^f - 8^V = 4(^2 + ^2) (^ + 2^v 



or 



fi - 2c*t^ - c\i2(T + 8/^) + cr*~ 4r"-((r + ir^) = o, 



NO. 1229, VOL. 48] 



a quintic for determining c, the approximation to Roche's limit. 

 If the two spheres are infinitely small compared with the sinele 

 one, this reduces to 



(^ = 12(7. 



Thus the factor 16 (which, as Mr. Heath shows, should have 

 been 8) of " G. R.'s" and of my previous letter must be 

 replaced by 12, when the rotation is taken into account. In 

 the notation used before, we therefore have as the approxima- 

 tion to Roche's limit 



2.29R . (5)1 



Proceeding further, as I did before, to find when three homo- 

 geneous spheres are in contact, so that <r = i and c = 2r + i 

 we have — ' 



22r'' - 2Sr* - 6or* + 14;^ -f 38 -I- 1 1 = o. 



Unity is a solution of this, so that three equal spheres are 

 in contact — an obviously correct solution. 



There is another root with r = 2-08, so that the two spheres 

 are each much larger than the third. 



These solutions of course give no approximation to that of 

 the problem to which the latter part of my letter referred 



May 3- G. H. Darwin. 



The Use of Ants to Aphides and Coccidse. 



^^ Mr. Cockerell is not quite accurate in saying that I have 

 "adduced the production of honey-dew by aphides as a diffi- 

 culty in the way of the Darwinian theory " (Nature, vol. xlvii. 

 p. 608). In the passage to which he alludes I have said, that 

 the relationship which in this matter subsists between ants and 

 aphides is one of the very few instances where it can be so 

 much as suggested that the structures or instincts of one species 

 have exclusive reference to the needs of any other species. 

 Therefore, even if this suggestion were not thus opposed to all 

 the analogies of organic nature, "most of us wonH protiably 

 deem it prudent to hold that the secretion must primarily be of 

 some use to the aphis itself, although the matter has not been 

 sufficiently investigated to inform us of what this use is" 

 (" Darwin and after Darwin," p. 292). 



But my object in now writing is to corroborate Mr. Cockerell's 

 explanation. For, on looking up my references, I find a letter 

 from the Rev. W. G. Proudfoot, dated March 26, 1891, in which 

 he communicates the following observations: — 



"On looking up I noticed that hundreds of large black ants 

 were going up and down the tree, and then I saw the aphides. 

 . . . Hut what struck me most was that the aphides showered 

 down their excretions independently of the ants' solicitations, 

 while at other times I noticed that an ant would approach an 

 aphis without getting anything, and would then go to another. 

 I was struck with this, because I remembered Mr. Darwin's 

 inability to make the aphides yield their secretion after many 

 experiments. A large number of hornets were flying about the 

 tree, but seemed afraid of the ants ; for when they attempted to 

 alight, an ant would at once rush to the spot, and the hornet 

 would get out of its way." 



From this it seems prohablethat, but for the preence of the 

 ants, the aphides would have been devoured by the hornets. It 

 also appears that Darwin's explanation is likewise true, viz. 

 that the aphides are bound to get rid of their excretions in any 

 case, and therefore that "they do not excrete solely for the 

 benefit of the ants." GEORGE J. Romanes. 



Christ Church, Oxford, May 6. 



Mr. Cockerell's letter (Nature, vol. xlvii. p. 608) suggests 

 the possibiliiy that the following fact bearing on the connection 

 between a coccid and another niemberof the Aculeate Hynieno- « 

 ptera may be interesting. ] have a quantity of Cotoneaster micro- I 

 phvlla covering a long sunny bank, and this shrub is much in- 1 

 fesied \iy ?i cocoXA, Secanium ribis. The queen wasps (usually 

 early in June, but this year ihev are beginning now) are attracted 

 in great numbers by the secretion from the coccid and may be 

 taken with a common ring net and destroyed, to the great advan- 

 tage of my garden. As to the visits of the wasps being of any 

 advantage to the coccid I am somewhat sceptical, though no 

 doubt they are to the wasps — when they are not caught ! 



Alfred O. Walker. 



Nant y Glyn, Colwyn Bay, May 5. 



i 



