138 BRIDGMAN. 



"microphone action" constitutes the departure from Ohm's law for 

 which we are searching. Various tacit assumptions will be made in 

 the course of this discussion which will be justified later. 



Return to Figure 1 for the bridge, and consider the heating effect 

 in the arm .r, treating its resistance as ohmic. The heat input is pro- 

 portional to (/i + ii cos uty, where ii is small compared with Ii. 

 Expanding this, neglecting the term in ir, the rate of heat input is 

 proportional to 7i^ + 2Zi ii cos ict, that is, there is a constant input 

 proportional to h^, independent of the presence of the A.C., and there 

 is a sinusoidal heating and cooling of the same period as the A.C. 

 which is proportional in intensity to both the D.C. and the A.C. 

 Under this heat input the conductor experiences a change of tem- 

 perature, which may be analyzed into a constant change dependent 

 only on the D.C, and a small alternating rise and fall, of the same 

 period as the A.C, but not necessarily in phase with it. The factor of 

 proportionaHty which determines the amplitude of the alternating part 

 is not the same as that which determines the amplitude of the steady 

 part, but is a function of the period, becoining less for higher fre- 

 quencies. Let us call the steady change of temperature to, the ampli- 

 tude of the in-phase part of the alternating part ti, and that of the 

 out-of-phase part t2. If the heat input is removed rapidly, to will be 

 small compared with ri. The increase of temperature above that of 

 the surroundings is therefore to + ri cos oit + t2 sin cat. Now if Ro 

 is the initial resistance at the temperature of the surroundings, a the 

 temperature coefficient of resistance, and R the actual resistance 

 when the current is passing, we have 



R = /?o [l + a (to + Ti cos o}t + T2 sin ut)]. 



The potential difference across the terminals of x is 



R(li + ii cos CO i). 



Expanding this by substituting the value of R above, and using the 

 relations 



cos2 = i (1 + cos 29), 2 sin 6 cos 9= sin 29, 

 we get : 



Potential difference = Ro{li{l -j- ccto) + | iiari] 



+ Rol haTi + ri(l -f- ocTo) } cos wt 

 -\- RolhaTo} sin co< 

 + i?ol 2 ^"laTi} cos 2aj/ 

 + Ro{^ iiocT-i} sin 2coL 



