Prof. J. D. Everett on Resultant Tones. 205 



Hence the rate of escape is proportional to 

 Ca o +(C-a o )/(0)-{/(0)P. 



a is comparable with the maximum value of /(#), and C is 

 much greater ; hence a may be neglected in comparison 

 with C. 



Developing {(/(#)} 2 , we shall obtain a term 



2a 1 6 1 sin (md + e^ sin (nQ + e 2 ) 



= a 1 o 1 [GO${(n — m)0 + e 2 — € 1 }—cos{(n + m)d + 6 2 + 6 1 }], 



representing a difference-tone and a summation- tone. From 

 (C— CLo)f(0) we have the common fundamental 



(C— a )A sin#, or GA sin 0, 

 and the two primaries 



C«i sin (?n# + e x ) , Chi sin (w0 + e 2 ) . 



Suppose for simplicity that fli=6i, then, taking the amplitude 

 of each of the primaries as 1, the amplitude of the common 

 fundamental will be A/%, and the amplitudes of the summa- 

 tion-tone and difference-tone will each be %/C. 



When n— ra = l, the difference-tone coincides with the 

 fundamental, and their joint amplitude may be taken as the 

 square root of the sum of the squares of %/C and A/%. 



13. Professor Pucker and Mr. Edser in experiments i. and 

 ii. obtained the difference-tone 64 from five distinct com- 

 binations of primaries, 



256 & 320, 192 & 256, 320 & 384, 51-2 & 115-2, 96 & 160, 

 their ratios being 



4:5, 3:4, 5:6, 4:9, 3:5. 



The second combination appears to have given a stronger 

 effect than either the first or the third ; whence it would 

 appear that low frequencies are favourable to strong effects. 

 Nevertheless the fourth combination is mentioned as giving a 

 rather feebler effect than any one of the first three. This 

 confirms our conclusion that the difference-tone is weaker 

 when it is distinct from the common fundamental than when 

 it coincides with it. 



Experiment iii. was directed to testing for the presence 

 of the resultant 64 when the primaries were 256 and 576. 

 which are as 4 : 9. Their common fundamental is 64, and it 

 could not be detected. This may have been because the 

 pitch 576 was too high to give a good effect. Or the failure 

 may be an indication that A/% is decidedly smaller than %/C. 

 It would be interesting to repeat the experiment, employing 

 192 and 320 as the primaries. 



