ELECTROCARDIOGRAPHY 



331 



FIG. 8. Demonstration of the Helmholtz theorem. (See text.) 



FIG. 9. A charged disc (membrane) of area Q representing an electromotive force which is re- 

 corded by the galvanometer of fig. 8 in an amount relative to its projection Q' on a plane perpen- 

 dicular to the current flow of the lead field. 8 is the angle between the flow lines and the moment 

 m of the disc. This means that m is represented in the record only as the smaller voltage m'. (See 

 text. ) 



FIG. 10. Principle of "lead field." The flow lines of a current entering the body at two arbitrary 

 points. El and E^, are shown. If a dipole V, represented by its vector, is recorded with the same 

 electrodes, the recorded potential is proportional to the projection m' of the moment m on the flow 

 line penetrating the region of the dipole. 



In a more general formulation, if U is the emf of the 

 battery, and i the current generated by U and pene- 

 trating Q, we find 



U 



(3-0 



If r is the resistance in the recording system,] • r is the 

 actually measured part of the voltage V of Q: 



W 



jr = V-1- 



U 



(3-2) 



If the resistance r is high compared with the internal 

 resistance of the body (a condition existing in every 

 modern recording instrument), the expression U/r 

 is practically equal to the total amount of current 

 driven by the external battery U through the whole 

 body, io. Therefore: 



\V = V- - 



(3.3) 



If io = I, the formula is transformed into 



W = V • i (3.4) 



The interpretation of equation 3.4 leads to the 



following concept: i is the current, a percentage of the 

 total current ic, which penetrates the surface Q. If the 

 current lines of i run parallel to Q, no current pene- 

 trates the surface, i.e., no part of the voltage on Q is 

 picked up by the electrodes and recorded by the 

 galvanometer. If the current lines run perpendicular 

 to Q, the maximum current penetrates the surface, 

 and thus the galvanometer records a maximum of the 

 potential \'. This leads, in general, to the relation 

 indicated in figure 9, that the current i is proportional 

 to the projection of the surface Q on a plane per- 

 pendicular to the direction of the current flow. 

 Since V = 4 tt-iti (equation 2.1 ), tlie recorded poten- 

 tial difiTerence \V is proportional to the "dot"' (scalar) 

 product of the vectors m and i, if m represents the 

 dipole moment per unit area, and i the vector of the 

 local current strength. 



W = 47r ( in- i) (3.5) 



(The sign of the potential difference is neglected.) 

 For our purposes it is more suitable to replace the 

 moment m by Mi/q, so that 



'"■ = ^" (7 • ') = ^^ (^I' q) 



(3-6) 



