570 



NA TURE 



[October 13, 1898 



Of the first series Prof. Michelson says: "This series in- 

 creases with n until we = ir. Suppose therefore e = kirln, 

 where -^ is a small fraction. The series will now be nearly 

 equal to nt ~ kir, a finite quantity even if « = oo. 



" Hence the value of y in the immediate vicinity oi x = it is 

 not an isolated point y = o, but a straight line -y = nx." 



Of the second series he says that it "is nearly equal to n for 

 values of nt less than ^-jr." 



Neither of these statements is correct. The sum of the first 

 series can be proved to be ^ (tt - e) when e lies between o and 

 2ir, and - J (it + f) when e lies between o and - 2ir, and it is 

 zero when e = o. The sum of n terms of the second series does 

 not approach to any definite limit, as n is increased indefinitely ; 

 nor does the difference between the sum of this second series to 

 n terms and the number n tend to zero or any finite limit, but 

 the ratio of the sum to ii terms and the number n tends to 

 the definite limit zero as n is increased indefinitely. 



The processes employed are invalid. It is not the case 

 that the sum of an infinite series is the same as the sum of 

 its first n terms, however great n is taken. It is not legiti- 

 mate to sum an infinite series by stopping at some convenient 

 «th term. It is not legitimate to evaluate an expression for 

 a particular value of x, e.g. x = ir,hy putting x =■ ir + e and 

 passing to a limit ; to do so is to assume that the expression 

 represents a continuous function. It is not legitimate to equate 

 the differential coefficient of the sum of an infinite series to the 

 sum of the differential coefficients of its terms ; in particular 

 the series given as representing dyjdx in the example is not 

 convergent. 



Lastly, Prof. Michelson says " it is difficult to see the mean- 

 ing of the tangent if j)' were an isolated point." The tangent, 

 at a point, to a curve, representing a function, has of course no 

 meaning, unless the function has a differential coefficient, for 

 the value corresponding to the point ; and a function which 

 has a differential coefficient, for any value of a variable, is 

 continuous in the neighbourhood of that value. 



St. John's College, Cambridge, A. E. H. Love. 



October 7. 



Helium in the Atmosphere. 



The letter of Mr.«Baly in your issue of last week, corro- 

 borating the statement of Friedlander and Kayser that helium 

 is a constituent of the atmosphere, induces me to put on 

 record a further confirmation of the accuracy of this observation. 

 Having had the opportunity, on June 20 last, of examining 

 samples of the more volatile portions from liquid air, which had 

 been handed to me by Prof. Dewar, I had no difficulty in 

 seeing the lines of helium in them. Further, a sample of the 

 helium separated by Prof. Dewar from Bath gas (following the 

 discovery of Lord Rayleigh) undoubtedly contained the sub- 

 stance called neon. 



In giving these facts I am only confirming the observations of 

 Prof. Dewar given to me in letters accompanying the samples 

 of gas. William Crookes. 



October 11. 



Triplet Lightning Flash. 



At the suggestion of Lord Kelvin, I send you the enclosed 

 photograph of a triplet lightning flash which was taken during 

 a recent thunderstorm at Whitby, and under the following 

 conditions. 



The flash must have been about two miles distant (out at sea). 

 The focus of the camera lens was 8 inches ; the aperture, f/64 ; 

 the plate, Ilford Empress. The camera was not stationary, but 

 was purposely oscillated by hand. It was intended that its axis 

 should describe a circular cone, but from the photograph the path 

 appears to have been rather elliptical. Each revolution occu- 

 pied about 1/80 minute. From these rough data I estimate that 

 the three flashes followed each other with a frequency of about 

 30 to 35 per second. They are identical in shape, but the top 

 part of the lowest (left-hand) one is missing, and the bottom is 

 screened. On the negative the centre flash is rather weaker 

 than the other two. Each flash is sharply defined on the left 

 edge and somewhat hazy on the right edge, due probably to the 

 gradual cooling of the glowing gases, and showing that the 

 lowest (left-hand) flash is the first of the three. The photo- 

 graph also contains a faint image of a single flash. During this 

 thunderstorm two other plates were exposed under the same 

 conditions as the above, but no images were found on them. 



NO. 151 I, VOL. 58] 



Possibly the lightning was too far off, and the aperture too 

 small. 



In view of the importance of obtaining more definite information 

 about lightning, I would suggest that in the presence of a 

 thunderstorm photographs similar to the above should be taken. 

 Greater accuracy than was possible under the above conditions 

 could be attained by rigging up the following simple contrivance. 

 An ordinary bedroom looking-glass should be placed on a table 

 in front of an open window facing the storm. The mirror should 

 be inclined at any angle of 45°. The camera tripod, with its 

 legs spread as wide apart as possible, should be placed on the 

 table so that its centre is over the looking-glass. The camera, 

 with its objective downwards, should be suspended from this 

 centre by means of three strings, and should be made to swing 

 in a circle by a gentle finger pressure close to the point of 

 suspension. The period of revolution should be noted. Should 

 any multiple flash imprint itself on the negative, it will now be 

 possible to accurately measure the intervals of time, except 



under the following conditions. If there are only two flashes, 

 the radius of the circle described by the camera can only be 

 guessed at. If the camera has described an ellipse, at least four 

 lightning images are required to find its elements. A camera 

 revolving on an axis passing through the objecti%'e would in 

 some respects be more convenient to work with, but unless it is 

 revolved by clock-work the time measurements would not be trust- 

 worthy. The aperture used by me, f/64, is probably too small 

 except for very brilliant flashes ; but if it is intended to allow 

 several discharges to imprint themselves on one negative, a very 

 large aperture will be found inconvenient because of the illumin- 

 ation of the landscape. The size of the aperture, rapidity of 

 plate, and distance of each lightning flash should be noted to 

 assist at forming some idea as to the heat generated. 



C. E. Stromeyer. 

 Lancefield, West Didsbury, October 3. 



The Centipede-Whale. 



The " Scolopendrous Millipede," which forms the subject for 

 the epigrams of Theodoridas and Antipater, and to which Mr. 

 W. F. Sinclair kindly called my attention (Nature, vol. Ivi. 

 p. 470), seems to mean a being quite different from the " Centi- 

 pede-Whale " which ^-Elian and Kaibara describe (see my letter, 

 ibid., p. 445) ; for the former apparently points to a huge 

 skeleton of some marine animal, while the latter is an erroneous 

 but vivid portrait of an animal actively swimming with numerous 

 fins. 



Major R. G. Macgregor, in his translation of the Greek 

 Anthology (1864, p 265), remarks upon the "Scolopendrous 

 Millipede " that the " word millipede must be understood rather 

 in reference to the extreme length of the monster than to the 

 number of its feet." However, it would appear more likely 

 that, in this similitude of the animal remains to the Myriapod, 

 the numerous articulations of the vertebral column as well as its 

 length played a principal part, should we take for comparison the 

 following description of an analogous case from a Chinese work 

 (Li Shih, " Stih-poh-wuh-chi," written thirteenth century, Jap. 

 ed., 1683, tom. X. fol. 6, b.) :— " Li Mien, a high officer (ninth 

 century), during his stay in Pien-Chau, came in possession 

 of one joint of a monstrous bone, capable of the use as ink- 



