340 DR. kcenig's researches on 



negative, to be found by dividing the frequency of the higher tone by 

 that of the lower. 



This mode of stating the facts is a little strange to those trained in 

 English modes of expressing arithmetical calculations, but an example 

 or two will make it plain. Let there be as the two primary sounds two 

 low tones having the respective frequencies of 40 vibrations and 74 vi- 

 brations. What are the two remainders, positive and negative, which 

 result from dividing the higher number, 74, by the lower number 40? 

 Our English way of stating it is to say that 40 goes into 74 once and 

 leaves a (positive) remainder of 34 over. But it is e<iually correct to say 

 that 40 goes into 74 twice all but G, or that there is a negative remainder 

 of 6. Well, Dr. Ktenig finds that, when these two tuning forks are 

 tried, the ear can distinguish two sets of beats, one rapid, at 34 per sec- 

 ond, and one slow, at 6 per second. 



Again, if the forks chosen are of frequencies 100 and 512, we may 

 calculate thus : 100 goes into 512 five times, plus 12 ; or 100 goes into 

 512 six times, minus 88. In this actual case the 12 beats belonging to 

 the inferior set would be well heard ; the 88 beats belonging to the su- 

 perior set would probably be almost indistinguishable. As a rule, the 

 inferior beat is heard best when its number is less than half the frequency 

 of the lower jirimary, whilst, when its number is f/reatcr, the superior 

 beat is then better heard. Ur. K^nig has never been able to hear any 

 lirimary beat which did not fall within this rule. 



Dr. Kcenig will now illustrate to you the beats, inferior and superior, 

 as produced by these two massive tuning-forks,* each weighing about 

 50 pounds and each provided with a large resonating cavity consisting 

 of a metal cylinder about 4 feet long, fitted with an adjustable piston. 

 One of them is tuned to the note «^i=64. The other also sounds ut^ ; 

 but, by sliding down its prongs the arijustable weights of gun metal and 

 screwing in the piston of the resonator, its pitch can be raised a whole 

 tone to ?-ei=72. Dr. KcFiiig excites them with the 'cello bow, first sep- 

 arately, that you may hear their individual tones, then together. At 

 once you hear an intolerable beating, the beats coming 8 per second. 

 This is the inferior beat, corresponding to the positive remainder ; the 

 superior beat you cannot hear. Dr. Kd'uig will raise the note of the 

 second fork from rci to mii=8i), and the beats quicken to 16 per second. 

 Raising it to/ai=85^, and then to soli=9G, while the first fork is still 

 kept at uti, the beats increase in rapidity, but are fainter in distinct- 

 ness. If Dr. Kcenig now substitutes for the second fork one tuned to 

 Zair=10G§, you may be able to hear two beats, the inferior one rapid and 

 faint at 42§ per second, and the superior one slower, but also faint, at 

 21^ per second. Still raising the pitch to the true seventh tone=112, 

 the rapid inferior beat has died out, but now you hear the superior 



* These apleuclid forlss, with their resonators, aloug with other important pieces of 

 Dr. Koeuig's apparatus, have since been acquired by the Science and Art Depart- 

 ineut for the Science Collection at South Kensington, 



