468 



NATURE 



[March 15, 1900 



means of the Fourier theorem, were therefore four times as 

 large as the curves here shown. 



I subjoin herewith a few specimen analyses. In the table, 

 column I. contains the vibration number of the fundamental, 

 i.e. the pitch at which the vowel was sung, and its amplitude 

 percentage ; column II. contains the vibration number of the 

 first overtone, i.e. the octave of the fundamental, and its ampli- 

 tude percentage, and so on. 



I 

 113 

 9*9 

 144 

 35l"9 



226 

 9-2 



II 



226 

 I5"4 



288 

 4*3 



IV 



452 

 o'8 



576 



684 

 17-5 



V 



565 

 11-5 



855 



VI 



678 

 161 



864 



1026 

 331 



VIIl 



904 



IX 



21-2 



152 1296 

 7-2 



X 



1130 



13-5 



1368 



XI 



1243 

 32 



1584 



3-6 34'6 80 31" J 2-6 



130 1356 1582 18 



1539 1710 

 6"i f9 



2034 2260 



It will be sufficient here to state as briefly as possible the 

 conclusions which I believe to be warranted. The vowels as 

 produced by the human organs of speech are composed in the 

 first place of two elements, that due to the vibration of the 

 vocal chords and that due to the resonance of the mouth, throat 

 and nose cavities. It is not always possible to separate clearly 

 these two elements, but at the pitches shown in the above table 

 the problem is quite simple for the vowel a. The fundamental 

 is due to the vocal chords, and the overtones that are strongly 

 reinforced are due to the mouth and throat resonance. The 

 vowel a, at any pitch, and pronounced by any clear voice, 

 contains the following partial tones : — 



(i) The fundamental, with the first two or three overtones. 

 The fundamental varies greatly in relative amplitude for reasons 

 which I do not as yet attempt to formulate. The overtones 

 are all weak, unless reinforced by the mouth resonance as set 

 forth below. 



(2) The overtone or overtones whose frequencies of vibration 

 chance to fall between looo and 1300 vibrations to the second, 

 the maximum seeming to lie at about 11 50. This is the main 

 characteristic of a, which serves to identify it to the ear, and 

 remains remarkably constant, no matter what the fundamental 

 may be. If the fundamental has 144 vibrations to the second, 

 overtones VII., VIII, and IX., with frequencies of 1008, 1152 

 and 1296 respectively, will all be present, but VIIL, with 1x52 

 vibrations to the second, in much the largest amplitude. 



(3) The overtone or overtones whose frequencies of vibration 

 chance to fall between 575 and 800 vibrations to the second for 

 mens voices, with a maximum at about 675 ; or between 675 

 and 900 with a maximum at about 800, for the voices of women 

 and children. This is presumably the resonance of mouth and 

 throat cavities resounding as one vessel, and is not as constant 

 as the main resonance described above. If the fundamental 

 has 144 vibrations to the second, we shall therefore find over- 

 tones IV. and V. present, with frequencies of 576 and 720 

 respectively, but V., being nearer the point of maximum reson- 

 ance, will be the stronger. 



The two regions of resonance are indicated in the table by 

 printing in larger figures the amplitude percentages of those 

 overtones which fall at or near the points of maximum reso- 

 nance, and are therefore present with large amplitudes. My 

 analyses reveal for the vowel a no other region of resonance 

 that is constant or important except these two. On the basis of 

 many observations it would be possible to plot a curve which 

 would represent the mouth resonance for the a-position. A 

 tentative chart of this sort will be found in the articles already 

 cited, and in general for a fuller discussion of the whole subject 

 the reader is referred to those articles. Louis Bevier. 



Rutgers College, New Brunswick, N.J. 



The number of works on natural history which are privately 

 published in England seems to point to a want of enterprise 

 among English publishers which I cannot understand, as the 

 market for really well illustrated* works on ornithology, botany 

 and entomology is certainly increasing. Any suggestions 

 addressed to me will be gratefully received. H. J, Elwes. 

 Colesborne, Andoversford R.S.O., Gloucestershire. 



Illustrations of Lepidoptera. 



Can any of your readers assist me in finding some artist who 

 is really competent to produce, by any process which combines 

 extreme accuracy with a reasonable cost, a large series of 

 illustrations of variation in butterflies ? The difficulty of getting 

 such work done in the country under my own eye is very great, 

 and I am unwilling to do as some of my friends advise, and get 

 them made in Germany. Where the variation is a question of 

 pattern only, photography would probably be the most satis- 

 factory process ; but where colour is the leading feature of the 

 variation, chromo-lithography seems the most likely to succeed. 



NO. T585, VOL. 61 I 



" Billiards Mathenaatically Treated." 

 The review, in your issue of March i, of Mr. Hemming's lx)ok 

 on billiards reminds me that I have never yet seen a satisfactory 

 explanation of the following question : — 



Why does a billiard ball when struck to the right or left of 

 its vertical axis nevertheless travel in the direction in which the 

 cue travels ? 



It only does so when the cue tip is chalked, otherwise it 

 travels in a direction of a line through the centre of the ball and 

 the point of contact with the top of the cue, as may be expected 

 from the laws of dynamics. The chalking of the cue of course 

 enables the striker to put on what is known as side, i.e. to make 

 the ball revolve on its vertical axis ; but why does it also allow 

 the ball to travel in a straight line instead of going off at an 

 angle? Enquirer. 



The answer to " Enquirer's " puzzle is very simple. If you let 

 a perfectly hard and smooth cue tip impinge at any angle on a 

 perfectly hard and smooth ball, the only force exerted will be 

 normal, and the ball will start in the line from the point of 

 impact to the centre of the ball. At the same time, during the 

 brief duration of the impact the cue tip will slide a minute 

 distance along the surface of the ball, but in consequence of the 

 perfect smoothness will exert no tangential force on the ball. 



Now take the extreme case in the opposite direction. Let the 

 cue tip be such that it bites on the ball without any sliding 

 whatever during the impact. In this case the point of the ball 

 struck moves exactly in the direction of the motion of the cue 

 — the centre of gravity of the ball moves in precisely the same 

 direction with a rotation added which is familiar to us as side — 

 and this will be so (if the bite is perfect), whatever may be the 

 angle between the direction of the cue and the surface of the 

 ball. With the ivory tips in use more than a century ago, you 

 had something approaching, though not quite reaching, perfect 

 smoothness — and the ball flew off not quite, but very nearly, in 

 the normal direction. With modern tips in good order and 

 well chalked the bite is sufficient to prevent any slipping and 

 drive the ball in the line of the cue, provided the angle between 

 the cue and the surface of the ball«s not too small ; and if this 

 condition is fulfilled the ball will start in a direction exactly 

 parallel to the line of the cue. 



If the angle is too small, as by playing side too near the 

 edge of the ball, there will be some slip and the ball will go 

 wrong (not quite normally, but along way from the desired 

 direction). It will be a miss-cue. The nearness to the edge 

 which can be played without miss-cueing is different for different 

 players, and depends upon some nicety in the handling of the 

 cue which does not admit of definition. 



Roberts can play side vastly nearer to the edge of the ball 

 than I should dream of attempting, and I daresay a good deal 

 nearer than " Enquirer" could try with success. But the broad 

 fact is, that with a well chalked cue you can insure (within 

 certain limits of angle) that there shall be no slip. 



If the cue is not chalked you can still do the same thing 

 within very much narrower limits, that is to say, you can put on 

 a very little side. 



The ivory tip, the chalked tip, and the unchalked tip, are 

 three different cases intermediate between the extreme theoretical 

 cases of perfect smoothness with unresisted slip and perfect bite 

 with no slip at any angle however small. 



I do not think that any one could have foreseen that a cue 

 tip could be made to bite to the extent to which it does. But 

 somebody in America or France found it out by experiment, and 

 for the first time made billiards the scientific game which it has 

 now become. . G. W. Hemminc. 



The Micro-Organism of Faulty Rum. 



In the course of our investigations upon this organism, first 



alluded to by us (Nature, vol. Ivi. p. 197, June 1897), and 



subsequently by others (vol. lix. p. 339, February 1899), we 



have found that from spirit of 70 per cent, of alcohol, which has 



