408 Prof. W. Cassie on the 



and the frequency is given by 



47r 2 w 2_ %\d 2 gx 



2 ~ (k x 2 + x 2 )W k, 2 + X 2 ' 



Likewise the frequency svith the wires 2d apart is given by 

 4?r 2^ 2 _ 2 *^ 2 + 0* 



Therefore 



tf W-"^ = (iSr • • • <*> 



The periods of the bifilar oscillation are the same as in the 

 first case. So that equations (1) and (4) give 



x -*^^( l+ 5> • • • • (5) 



as # is always small compared to \, x 2 jk 2 is usually 

 negligible, and when this is so equation (5) reduces to (3). 



Thirdly, to allow for the hooks supporting the knife-edges, 

 let m be the sum of their masses, and k z their radius of 

 gyration about a vertical axis through their common centre 

 of gravity. Then if rc 4 and nj are the frequencies of pitching, 

 and n z and nj of bifilar oscillation with the wires distant 2c 

 and 2d apart respectively, we have 



toM-** ' M+m + 2 -l— 



^ b ~ I 'Mkf + mkf m^ + mkf 



2 2 _2Xc 2 1 



and corresponding equations with d substituted for c. 

 Thus we get 



since m is small compared with M, and c, of, and & 3 are small 

 compared with k v When the fraction in the last bracket is 

 negligible, or when the hooks are so shaped that k B = sj c 2 + d 2 , 

 equation (6) reduces to the original result of (3). 



II. Second Oscillation Method. 



In certain cases it is necessary to clamp the wires direct to 

 the needle instead of attaching them to hooks which support 

 the needle by knife-edges. In this case flexure of the wires 

 has to be taken account of. 



