2 20 



NA TURE 



[August 19, 1909 



colleagues, who are at present absent in the lield, but we 

 are convinced that if they had the opportunity they 

 would join with us in appending their signatures to this 

 letter. 



T. H. D. La Touche, H. H. Hayden (Superintendents, 

 Geological Survey of India). 



P. N. Datta, E. Vredenburg, L. L. Fermor, G. E. 

 Pilgrim, G. H. Tipper, H. Wa'lker, K. A. K. Hallowes, 

 G. de P. Cotter, J. J. .\. Page, H. C. Jones, A. M. Heron, 

 M. Stuart, N. D. Daru (."Assistant Superintendents, Geo- 

 logical Survey of India). 



W. A. K. Christie (Chemist, Geological Survey of 

 India). 



Geological Survey of India, Calcutta. 



An Optical Phenomenon. 



In reference to the query of "V. P." in Nature of 

 June 3 (p. 398), under the above title, I describe a simple 

 experiment which will, in all probability, lead to an easy 

 explanation. 



Allow sunlight to fall upon a vessel filled with water to 

 a depth of a few inches. If the bottom be white, so much 

 the better. A bath-tub is excellent. Now draw a finger 

 through the water so as to produce a wake, in which are 

 to be seen the familiar " dimples " characteristic of vortex 

 motion. Then, on the bottom will be observed, correspond- 

 ing to each dimple, a black shadow with a brilliant edge, 

 just the same sort of appearance, in fact, as that described 

 by your correspondent. The same, by the way, may be 

 observed in shallow brooks. 



The explanation in this case is not difficult. A very small 

 central portion forms a concave lens, the enfeebled illumina- 

 tion of which on the bottom is negligible. The portion sur- 

 rounding this and extending as far as the plane water- 

 surface acts somewhat after the manner of a convex lens, 

 concentrating the light passing through it into a more or 

 less sharply defined ring, a. " focal ring," so to speak, as 

 contrasted with a " focal point." The diameter of this 

 ring would approximate that of the whole dimple. By far 

 the greater part of the light falling upon the area of the 

 dimple is collected here, and, consequently, the field within 

 appears black aided by contrast. It is easy to see that an 

 essential is the relative smallness of the concave part of 

 the dimple, as is borne out by failure to obtain the 

 phenomenon on a large scale by stirring water in a beaker. 



Now, following up this e.xperiment and considering 

 window-paies, one would expect to find there flaws of a 

 dimpled nature, or else of a corresponding heterogeneity 

 of refractive index. The former I have found to be the 

 case, especially surrounding air-bubbles, as is easily to 

 be detected by the touch in many cases. The formation 

 of these flaws could be accounted for by the contraction of 

 the air during cooling in process of manufacture. More- 

 over, in this instance, the bubble itself, forming the concave 

 lens, need not always be small, since it is usually of a 

 focal length far shorter than that of the surrounding portions 

 of the pane. 



Flaws of this type are rarer than those of an opposite 

 or protruding type, which, of course, produce patterns with 

 a white centre. L. G. Hoxton. 



University of Virginia, July 26. 



A Quest'on of Percentages 



In reply to the letter on the " Calculation of Percent- 

 ages " in Nature of August 5, may I venture the opinion 

 that the only common-sense method of finding the per- 

 centage of marks gained by a student in a series of ex- 

 aminations is to add together all the marks obtained by 

 the student, and find what percentage this total is of the 

 maximum possible? 



By this method the more difficult papers, which have a 

 greater number of marks allotted to them, retain an 

 important proportion in the result, whereas the elementary 

 papers, which are worth only a few marks, have only 

 a small influence on the final percentage. 



If one were to calculate the percentage for each paper, 

 and then average these percentages, this would manifestly 

 be the same as giving the same number (i.e. 100) of marks 

 for each paper. This would necessitate equal difficulty in 

 each paper set. Lewis \\'hallev. 



39 Clarendon Street, Keighlcy, Yorks, August 13. 

 NO. 2077, VOL. 81] 



In Nature of .August 5 Mr. Cunningham appeals to 

 mathematical readers for information on the question of 

 averaging three results, viz. : — 



37/5o+5o/5o+7>/ioo. 

 either giving 



(37+50+70/200 = 79/100 

 or 



(2 X37-I-2 xso-|-7i)/3oo = 8i§/ioo. 

 Though I am not a mathematician, but a chemist, I trust 

 I can give the required answer. 



The way of averaging depends on the weight of the 

 single results. By the latter way of calculating, the third 

 result affects the average with twice the weight of the 

 former way. Equal weights for each result require equal 

 denominators. Taking for a very simple instance the 

 problem of averaging between 20/100 and 40/100, which 

 is obviously 30/100, the first way of averaging, as proposed 

 by Mr. Cunningham, would permit a calculation like this 

 (20/100 being = 2/io= 1/5, &c.) : — 



2/10-1-40/100 = 42/110 or 1/5-1-40/100 = 41/105, 

 or (40/100 being = 40o/iooo = 400o/ 10,000) 

 1/5-1-400/1000 = 401/1005 or 1/5-1-4000/10, 000 = 4001/10, 0C5. 



These results, I believe, will explain better than many 

 words the essential point of this question. 



Breslau, Parkstr. 13. R. Abegg. 



Kohlrausch's " Physcal Measurements." 



Replying to the letter of Mr. Nelson in Nature of 

 .\ugust 12, the value given for the fe in question in the 

 ninth German edition of " Kohlrausch's Lehrbuch der 

 praktischen Physik " (1901) is 0-457. But it must be 

 borne in mind that this value is deduced on the assump- 

 tion that the specific gravity of the brass weights is 84, 

 and seeing that the specific gravity of various samples of 

 " brass " varies not inconsiderably, it is immaterial 

 whether one uses 0-457 or 0458 for the correction factor. 

 l"he rounded value, 0-46, is near enough for most pur- 

 poses, and that is the one given in the tables of Landolt- 

 Birnstein. For accurate work the specific gravity of thp 

 weights must be determined in any case, and the value 

 of h calculated for these particular weights. 



G. Rudorf. 



" Ivor," Cranley Gardens, Muswell Hill, 

 London, N., August 12. 



A Kinematical Illusion. 



The following experiment is easily tried, and throws, 1 

 think, some light on a certain type of illusions. 



A small cogwheel from an old American clock is the onlr 

 apparatus required. Holding the axle in the finpfer and 

 thumb of the right hand, give it a twirling motion, saj 

 counter-clockwise. Let the teeth of the wheel click gently 

 against a small card, or the finger-nail of the left hand. 

 On looking at the wheel the spokes appear to revolve 

 counter-clockwise (as they do) and the teeth to revolve in 

 the reverse direction. C. S. Jackson. 



25 Nightingale Place, Woolwich. 



RONTGEN RAYS IN THE DIAGNOSIS OF 

 DISEASE. 



GRE.\T development has taken place in the last 

 few years in the application of Rontgen rays to 

 the diagnosis of disease. At first it was only possible 

 to show the shadows cast by bones and bv dense 

 foreign bodies, usually metallic bodies. With im- 

 provement in apparatus and in method, the art of 

 radiography has advanced in such a way that it is 

 now possible to show, not only the outlines of the 

 bones, but minute details of their structure, and, more 

 than this, a considerable amount of detail can now be 

 shown in the soft parts of the limbs. While at first 

 surgeons alone found X-ray diagnosis useful, as in 

 the diagnosis of fractures, dislocations, and foreign 

 bodies, the physicians have gradually been able to 



