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 x, treating its resistance as ohmic. The heat input is pro- 

 portional to (/i + ^1 cos caty, where ii is small compared with 7i. 

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

 proportional to /i^ + 2/i ii cos cot, that is, there is a constant input 

 proportional to 7i^, 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 wath it. The factor of 

 proportionality 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, becoming less for higher fre- 

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

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

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

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

 the surroundings is therefore to + n cos cot -\- t2 sin cot. Now if Rq 

 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 = i?o [l + a (to + Ti cos cot + T2 sin wt)]. 



The potential difference across the terminals of x is 



R{li + ii cosoj^). 



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

 relations 



cos^ = i (1 + cos 26), 2 sin 6 cos 6= sin 2d, 

 we get : 



Potential difference = i?o( A(l + arc) + I hcxTi} 



-f RolhaTi + fi(l + q:to)} COS wt 

 + Ro{liaT2} sin cot 

 + i?ol| ^i«Ti| cos 2cof 

 + Ro\^ iiocTo} sin 2cot. 



