154 The Ohio Journal of Science [Vol. XVIII, No. 5, 



Whence for two different distances .ri and ,to we can say 

 7i = l^Q-""^ and /o = Le""'-^ whence 



loge /1//2 



a 



.X-o-.Vi 



Now for the third harmonic .r2 = ^^ and .ti = -^^ whence 



4X3 1 ,. 2X3 



.T2 — .Ti = X3. Similarly for the fifths and sevenths .vo — .Ti would 

 equal X5 and X- respectively. The ratio — ^ for the thirds is 



= 1.099. For the fifths and sevenths this ratio is 



101 



^ = 1.044 and 114 = 1-017 respectively. 

 36.9 11.2 



Using the figures for Xg given for 7/ = 8 cm. viz., X3=145, 

 X, = 86, X7 = 61, we get a to be 0.00065, 0.00050, 0.00028 respec- 

 tively. Thus the value of a apparently decreases for the higher 

 harmonics. It may fairly be questioned, however, whether 

 the question of damping can be treated as simply as here given. 

 Using the values of Xs corresponding to the optimum values of y, 

 which we shall call y^, viz., X3=150, Xs^^S? and X7 = 61, changes 

 the values of a but little. Now Blake and Sheard* have shown 

 that the electrostatic leakage, which they called "^ (z/) " was 

 greater for the lower tones than for the higher tones. This 

 would readily explain the larger values of a for the lower 

 tones. 



The important thing at present, however, is that in any case 

 the value of a must be very small. Even though we measured 

 X, not from the oscillator spark-gap, but from the Lecher plates, 

 the value of a would be doubled only. There seems no good 

 justification for doing this, however. 



For the third harmonic, using the value of a first given, 

 0.00065, and calculating the intensity / of the third at the 

 middle of the Lecher system we get 7=105.9. The arithmetic 

 mean of 111 and 101 is 106.0. This is the justification for 

 taking the arithmetic mean of the two tones either side of the 

 fundamental to get the tone intensity of any tone that can 

 fairly be compared with the fundamental. The question of 

 damping is thereby eliminated. 



Physical Review, 1. c. 



