21 G On the Bridge Metliod and Periodic Currents. [Feb. I 1 . 1 . 



inductance is TSxlO* C.G.S. Again, if C be one microfarad, equal 

 to 10- M C.G.S., R is 160 ohms, and L ia 2'5 x 10 7 cm. 



In the preceding calculations e and / are supposed to be adjusted 

 to the values most favourable to the effect in the receiving instru- 

 ment. A question, which arises quite as often in practice, is how to 

 make the best of given instruments. The full answer is necessarily 

 somewhat complicated ; for there could be no objection to the inser- 

 tion of a condenser for example, if the sensitiveness could be im- 

 proved thereby. In what follows, however, the transmitting and 

 receiving branches will be supposed to be fully given, so that e and / 

 are known complex quantities ; and the only question to be considered 

 is as to the most suitable value of a, assumed to be equal to c. 



For this purpose the modulus of the second fraction on the right 

 in (19) is to be a maximum, or that of 



is to be a minimum, by variation of a. The problem thus arising of 



determining the minimum modulus of a function of a complex 



quantity may be treated generally. 



Let 



F (z) = F (x + iy) = 



and let it be required to find when the modulus 2 of F (z), viz.,. 

 , is a minimum by variation of x,y. We have 



ax 

 And in general 



_ 

 dx dy 



+* = o .... (4*). 

 ay dy 



dtp _ dy- 

 Ty~ ~te 



In oi-der that (44), (45) may both obtain, we must have eit 

 0- + yr 2 = o, or else 



T~ ~T~ *"~ > ~j~ 



dx dy dx 



The latter conditions are equivalent to 



A 



dy 



For example, let 



F(z) = 



where a, ft are complex constants. 



(46). 



(47), 



