146 PROCEEDINGS OF THE AMERICAN ACADEMY. ^ 



that a becomes zero; hence the vakie of co, which in the experimental 

 circular graphs of Figures 7 to 10 lies at the remote end of the principal 

 diameter is the co = coq for which the diaphragm in sustained vibration 

 is resonant. This gives a simple and accurate method of determining 

 coq for a telephone diaphragm.* 



Again, let A be the logarithmic decrement per second of the dia- 

 phram, if vibrating under no external force, then by the theory of 

 elasticity, 



r 



whence from (36) 



A = — -, numeric /sec. (38) 



2m 



CO tan a = — ^ numenc/sec. \6v) 



2A 



Differentiating (39) with respect to a, we obtain 



dw . „ oi d<ji . . ,.^s 



-r- tan a + CO sec- a = — - ^- numenc/sec. (4UJ 

 da A da 



and if a = 0, 



[ 



-7- = A; numeric /sec. (41) 



daJo 



That is, in the experimental circular graphs, the rate of change of co 

 with change of a, at the remote end of the principal diameter, is the 

 logarithmic decrement per second of the diaphragm. This quantity 

 cannot, however, be obtained with the precision with which co can be 

 obtained. 



Another method of obtaining A is by taking the values of coi and 

 aj2 which lie respectively 45° below and 45° above the principle diame- 

 ter, — these angles being measured at the origin, not at the center. 

 For these points tan a is respectively + 1 and — 1 ; whence from (39) 



2Acoi = coi^ — coo^ 



2AC02 — C02^ COo^ 



and by subtraction and division by 2 (coi + coo), 



A ''^1 ^^2 • / /.n\ 



A = — numenc/sec. (42) 



4 For another method of finding wo from the humming tone of a telephone 

 receiver, see a paper by A. E. Kennelly and W. L. Upson, Proc. Am. Phil. Soc, 

 1908, "The Humming Telephone." 



