OHM ON' THE GALVANIC CIRCUIT. 413 



BC:PX' = x':a/, 



then 



-X'Y' = ^(x + ^)-a. 



Thirdly, since C D = K K' and F" X" is equal to the part of 

 K K' which extends from K to the line X" Y", -we have 



C D : L K' = P'X" : X" Y" - K F", 

 whence 



X" Y" = ±^^\-^^ + K F", 



or, since K F" = K I + I H'- F' H and F H = G H- G F, 



T T\ ' F' Y" 



X"Y" = ^jy + I H' + GF - (a + a'). 



If now for L K', I H', G F' we substitute their values 

 A . X" A . V A . X 



L ' L ' L 



we obtain 



X" Y" = ^ (a + X' + ^"^p ^") - (a + a') ; 



and if by x" we represent a line such that 



C D : F" X" = X" : x" 

 we have 



X" Y" = ^ (X + X' + ^') - (« + «')• 



These values of the ordinates, belonging to the three distinct 

 parts of the circuit and different in form from each other, may 

 be reduced as follows to a common expression. For if F is 

 taken as the origin of the abscissae, FX will be the abscissa 

 corresponding to the ordinate X Y which belongs to the ho- 

 mogeneous part AB of the ring, and x will represent the length 

 corresponding to this abscissa in the reduced proportion of 

 A B : X. In like manner F X' is the abscissa corresponding 

 to the ordinate X' Y' which is composed of the parts F F' and 

 W X' belonging to the homogeneous portions of the ring, and 

 X, a;' are the lengths reduced in the proportions of A B : X and 

 B C : x' corresponding to these parts. Lastly F X" is the ab- 

 scissa corresponding to the ordinate X" Y", which is composed 

 of the parts F F', F' F", F' X" belonging to the homogeneous 

 portions of the ring, and X, x', w" are the lengths reduced in the 

 proportions of A B : X, B C : x', C D : X". If in consequence 

 of this consideration we call the values x, X + x', \ + x' -\- x'' 



