166 Drs. Orehore and Squier on the Currents 



cidence at P. The condition that P and Q shall coincide, by 

 referring to the figure, and observing that the points H and 

 K lie upon a semicircle having OP as a diameter, while M 

 and N lie upon one having PA as a diameter, may be 

 expressed by the equations derived from the right triangles 

 OHPandOKP, viz.:— 



or 



Again 



OP 2 = OH 2 + HP 2 = OK 2 + KP 2 , 



I 1 2 3 (R 2 + L 2 ^) = I 2 2 4 (E 2 + L|^). . (3) 



PA 2 =PM a + MA 2 =PN 2 +NA 2 , 



I 2 3(R 2 + L> 2 ) = I 2 ^(P 2 + L 2 a, 2 ). . (4) 



Dividing equation (3) by (4), member for member, we 

 obtain 



or 



B2+-LS»* Ul+Ll^' 



(5) 



or denoting the impedances of the branches by J 1? J 2 , J 3 , and 

 J 4 respectively, and remembering that Ji = n/ R 2 + Li o> 2 • 

 J 2 =&c., we have 



T-T) ( 6 ) 



'J 3 »J 4 



as the necessary condition for zero current in the galvano- 

 meter. 



This equation asserts : 



When an harmonic electromotive force is impressed upon one 

 of the branches of a Wheat stone bridge, a galvanometer in the 

 conjugate branch of the bridge can only indicate zero current 

 when the impedances of the remaining four branches of the 

 bridge form a simple proportion. 



This is entirely analogous to the well-known condition 

 when the direct current is used in the Wheatstone bridge, 

 namely, that the resistances in the branches of the bridge 

 form a proportion. 



Another expression which the diagram makes apparent, 

 and will be useful to note, is that derived from the right 

 triangles OAO and OAC inscribed in the semicircle. These 



give 



OA 2 = OC 2 + CA 2 =OC' 2 + C /2 A 2 , 



or E 2 =I 1 2 , 3 [(R 1 + B 3 ) 2 + (Li + Ls) 2 *) 2 ]. 



= l| 4 [(R 2 + E 4 ) 2 + (L 2 + L 4 ) 2 co 2 ]. , . (7) 



