Formula used in Inductance Measurements. 803 



Now cos« = -, where p± is the total impedance of 



P l branch OAjB, 



and tana = rr , so o(tan«) = ■ _ . 



ri + IV n + Ki 



H^t^ ci^^twSL Wio>SL 

 ence SE = * = — L ^— . 



Pi Pi 



(2) Let Rj become Ri — SR 1? the inductances and all the 

 other resistances remaining at the balancing values. P again 

 moves in the same direction along the semicircle OP 2 B to 

 some point Pj (fig. 7), Let A/ be the corresponding posi- 

 tion of A so that A/A 2 represents the P.D. SE' across the 

 telephone. Since i\ and L x are unchanged, 



OV _ BP , _ fr+Sci 

 OA 2 - BP 2 ~ Cl ' 



Also angle P 1 OP 2 = angle PiBP 2 , so the triangles OA 2 A'! 

 and BPoPj are evidently similar, so that when ha is small 

 the angle OA 2 A/=7r — a. 



Hence we see that A 2 A X is perpendicular to A 2 A/, i. e. the 

 P.D. across the telephone due to a small difference between 



j- and — is in quadrature with that caused by a small 



-U 2 To 



difference between ^ and — . This fact greatly helps the 

 R2 r 2 



operation of balancings especially if our standard inductance 



is variable, as, even if — 1 has not exactly the right value, 



R2 

 the inductance ratio which gives the least sound is very 



nearly - . Similarly, if this adjustment is imperfect, the 



value of ~ which gives the least sound is — • Hence we 

 ±\ 2 r 2 



can make the final adjustments independently and succes- 

 sively, repeating the steps until the best balance is reached. 

 "We have 



£F'— ( h^ >0L — ClV{ cos a ' ^ (^ an a ) 

 sin « tan a 



